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Student_t

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The Feasibility of Omniscience
« on: June 08, 2013, 02:07:52 pm »
My post regards whether or not the definition of "God" (as used by Dr. Craig) is a proper definite description. Among other things, God's posited as a being which is omnipotent and omniscient. However, is there good reason to think that there are upper-bounds on power and knowledge, so as to allow this to be feasible?
   
Let's look at one area in which any being's abilities and knowledge is necessarily finite and incomplete. For example, it is obvious that there's no greatest natural number, as this set is an infinite one. Now, we know that no possible being can know all of the elements of this set, since it's uncountable. We can also say that if one being knows or can know more natural numbers than another, then that being is more knowledgeable or capable, respectively, in this area. This means that since the set of natural numbers is infinite, there cannot, in general, be a single greatest-possible being, because we can always conceive of a being with greater cognitive capabilities or knowledge (i.e, the ability to know n + 1 elements of the set). Since there are certain areas pertaining to knowledge and power which do not have an upper-bound, it is necessarily the case that any possible being's net knowledge and capabilities are limited and imperfect. Hence, there cannot be a being possessing the properties that are ascribed to God.
Try the following program:

def recursive_fun(s='noititsrepus si dog'):
   if len(s) == 0:
      return ''
   else:
      return recursive_fun(s[1:]) + s[0]

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John M

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Re: The Feasibility of Omniscience
« Reply #1 on: June 08, 2013, 06:13:30 pm »
I understand you are a new member, Student_t, so I would like to mention that it is considered poor form on this forum to initiate different threads with identical questions. Please only post the same question once rather than litter the forum with the same question multiple times.

For those interested, this question was posted and its discussion followed here.

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JTega6

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Re: The Feasibility of Omniscience
« Reply #2 on: October 31, 2018, 09:09:51 pm »

Your argument goes something like this:

If there are infinities, then there cannot be a single greatest possible being.
There are infinities.
Therefore, there cannot be a single greatest possible being.

My main objection would be to premise 1. I do not think that the presence of infinities makes the existence of a greatest possible being impossible. I think that the main point that the argument goes wrong is with the analogy. In your analogy you use the example of the natural set of numbers being infinite to show that there is no greatest possible number and use that narrative to show how there cannot be a greatest possible being. I think that where you are misguided is thinking that God resembles the greatest possible number, instead he would actually be the natural set of numbers. God’s attributes contains infinities just as the natural set of numbers does but God in himself exists just as the natural set of numbers does. His goodness, wisdom and power might be infinite as the number line increases but that does not mean he does not exist as a greatest possible being. If you were to try to imagine a being with an extra element of goodness or power than God, all you really did was just imagine God again except more thoroughly this time. Just as if you added another number to the natural set of numbers, it wouldn't disqualify the concept of the number line you previously had as the natural set of numbers, it just expounded on it. In this way it does not follow that the presence of infinities leads to the nonexistence of God, instead the existence of infinities makes it harder on us to fully grasp God’s true nature. Our inability to fully understand God does not show that God is logically impossible, if a greatest possible being did exist then it would follow that lesser beings would have a hard time fully comprehending the greatest possible being as he would be greater than they would be able to grasp.

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jayceeii

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Re: The Feasibility of Omniscience
« Reply #3 on: November 05, 2018, 02:50:15 pm »
I think that where you are misguided is thinking that God resembles the greatest possible number, instead he would actually be the natural set of numbers. ... Our inability to fully understand God does not show that God is logically impossible, if a greatest possible being did exist then it would follow that lesser beings would have a hard time fully comprehending the greatest possible being as he would be greater than they would be able to grasp.
JTega6,

The proper defense of God in this instance is not claiming infinities, but as you also state, God’s greatness is not comprehensible to the entities He made. The trouble with infinite sequences is that you can always find a next number, and God finds that too, no matter how far out it goes. The entire universe could be wrapped up in simply contemplating the natural numbers, not to mention the other infinite sets such as the Fibinocci sequence. If you want to say, “My God is greater than that,” you haven’t actually imagined a greater God. Instead it is a meaningless postulate, just like saying God would exist “outside of spacetime.” At first it sounds elegant and powerful, but you can’t explain what it means. Instead you might want to postulate God keeps tracks of all thoughts and movements of all humans and animals on the globe across vast eons. Then you’re talking about a possible but incomprehensible Deity. It is wrong to define “imperfect” as “incapable of absurdity,” as containing all natural numbers would be. God is Perfect, but not absurd.

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noncontingent

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Re: The Feasibility of Omniscience
« Reply #4 on: November 19, 2018, 02:51:27 pm »
What I don't get is that if God has a mind and can think, then how can he know his own thoughts as to his own omniscience aren't implanted by a higher order intelligence?

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JLouis

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Re: The Feasibility of Omniscience
« Reply #5 on: November 26, 2018, 09:13:03 pm »
@Student_t, I appreciate you declaring your bias; the program in your footer returns 'god is superstition'. I will reciprocate by declaring my bias; I know God through my personal relationship with Him.

...
This means that since the set of natural numbers is infinite, there cannot, in general, be a single greatest-possible being, because we can always conceive of a being with greater cognitive capabilities or knowledge (i.e, the ability to know n + 1 elements of the set).
...

Regarding omniscience, your argument seems to have the following structure.
1. If God is omniscient, then God's knowledge is at least as great as the knowledge of any conceivable being.
2. If a being knows finitely many natural number, then there is a conceivable being whose knowledge of natural numbers is greater than the being’s.
3. If a being knows infinitely many natural numbers, then the being's knowledge is infinite.
4. Every being's knowledge is finite.
5. God is a being.
6. (By 5 and 4) God's knowledge is finite.
7. (By 6 and 3) God knows finitely many natural numbers.
8. (By 7 and 2) There is a conceivable being whose knowledge of natural numbers is greater than God's.
9. (By 8 and 1) God is not omniscient.

This is a good structure for an argument. However, the premises are presented without justification and are subject to critique.

Premise 1 (and a little of premises 2 and 3).
This premise implicitly assumes there is an "is at least as great as" relation between any being's knowledge and another's. First, there is no obvious (to me) reason that knowledge is subject to such a relation. This assumption thus requires justification. In addition, the narrative demonstrates this relation is the "is a superset of" relation. However, invoking the notion of greatness is an appeal to an external standard. As this standard is unexplored here, there is no justification for accepting that the standard greatness equates the "is at least as great as" relation on knowledge with the "is a superset of" relation on knowledge. It is worth noting that some hold these relations to be equivalent because of their beliefs about the greatness of knowledge.

However, it is not clear that the language of sets and supersets applies to a being's knowledge at all. It is conceivable that the limitations of set theory are too restrictive to adequately describe or apply to a being's knowledge. Without the tools of set theory to guide the remainder of the original argument, premise one has no clear meaning and premise two is unjustified. It may or may not be true for whatever rules apply to relations between one being's knowledge and another's. Later, I will show this also brings premise three into question.

Premise 2 (and a little of premise 4).
Consider the hypothetical that God knows all possible thoughts of all actual and conceivable beings. First, if there are infinitely many actual or conceivable beings who each have possible or actual knowledge of who they are (as distinct from each other being), then God's knowledge would be infinite. This presumes the rules of set theory apply to a being's knowledge (more on this later).

Returning to the natural numbers, I can conceive of a being that knows the number one, and another being that knows the number two, and a third being that knows the number three, and so on. In this case, it might be convenient to say God would then know all natural numbers because He would know the thoughts of all these conceivable beings.

However, I have not actually conceived of all these beings. Contrariwise, there may be natural numbers so large that it would take me longer than my entire life to conceive the number because of how large it is. Since I must actually conceive the number in order to conceive of a being that knows the number, such a being is inconceivable by me. If there is any natural number so large that no being could conceive of it, then a being that knows this number is inconceivable. In this case all natural numbers larger than this inconceivable number are also inconceivable; hence there are only finitely many conceivable natural numbers.

Let us take this view for the moment (that there are finitely many conceivable natural numbers). Restricted to natural numbers, this would only require that all of the finitely many conceivable natural numbers are known by God. Thus, in this view, a being's knowledge of natural numbers can be both finite and greatest. This can be extended to all conceivable knowledge, not just conceivable knowledge about natural numbers. That is, the form of the argument applies to there being finitely many conceivable natural numbers and to there being finitely many conceivable elements of knowledge of any kind. Thus, in this view, a being's knowledge of everything can be both finite and greatest. In this case, premise two is false.

Now consider the view that there are no inconceivable natural numbers. Given our hypothetical (that God knows all possible thoughts of all actual and conceivable beings), we conclude that God knows all the natural numbers. In this case, premise four is false (slightly more on this premise later).

All of this can be attacked by attacking the initial hypothetical, that God knows all possible thoughts of all actual and conceivable beings. If this is false, then the rest of my counterargument fails. It is worth noting, though, that some hold a version of this hypothetical as the definition of omniscience. For such people, my counterargument is a valid critique of the original argument.

Premise 3.
Suppose the language of sets and supersets applies to a being's knowledge. Then the original argument holds that a being's knowledge is infinite if the cardinality of the being's knowledge is infinite. Premise three can then be split as follows. If a being knows infinitely many natural numbers, then a subset of the being's knowledge is infinite. If a subset of a being's knowledge is infinite, then the being's knowledge is infinite. In this case, the first part of the split follows from treating things a being knows as elements of the being's knowledge. That is, there is an injective function from the infinite set of natural numbers known by the being to the being's knowledge (treated as a set). Adding to this the formal rules of transfinite cardinality of sets, the cardinality of the being's knowledge is infinite. That is, we can conclude the second part of the split from the rules of set theory.

Now suppose the language of sets and supersets does not apply to a being's knowledge. Then premise three cannot be split as above, because the split is not supported by the rules of set theory in this case. In particular, there is no rule to help us specify any conditions at all under which a being's knowledge would be infinite. Could it be that a being knowing the set of all natural numbers is consistent with the being's knowledge being finite? There is no rule that produces a contradiction with this. Perhaps knowing the whole set only constitutes one element of knowledge.

This may sound strange, perhaps even absurd. However, the field of mathematics is replete with infinite sets specified all at once with one definition. The set of all prime numbers is the set of natural numbers that have exactly two divisors. The set of natural numbers is the set of entities that result from repeated applications of the successor function (using Peano's Axioms to formalize the set of natural numbers). The set of rational numbers is the set of ratios of the form a / b where a and b are integers and b is not zero. Do these sets defined by properties constitute three elements of knowledge or infinitely many elements of knowledge?

Of course, this approach can be scrutinized; and, it should be scrutinized. However, this can serve as a starting point for exploring the validity of premise three. It is unclear that the rules of set theory apply or ought to apply to a being's knowledge. This is unjustified in the original argument. If the rules do not apply, is there a coherent model of knowledge that is inconsistent with premise three? I would like to explore this more with a willing participant.

Premise 4.
This premise is presented in the narrative as being obviously true. However, there is no justification presented for it. It is logically independent of the other premises; that is, it does not follow from the other premises. And, I do not believe is it true. That is, I have no philosophical issue with the negation of premise four. Without justification of this premise, the original argument is unconvincing.

Premise 5.
I have no reason to doubt this premise. It follows directly from the definition of God.

Conclusion.
The OPs argument was described using the language of formal logic. Various reasons are given to doubt the applicability or validity of some premises of this argument. Other premises are simply unsupported. The result is that the original argument is unconvincing. I welcome critique of my response so long as it is made with a good faith effort to reach a common understanding.

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jayceeii

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Re: The Feasibility of Omniscience
« Reply #6 on: December 04, 2019, 02:51:28 pm »
Response to Jlouis, Part 1
@Student_t, I appreciate you declaring your bias; the program in your footer returns 'god is superstition'. I will reciprocate by declaring my bias; I know God through my personal relationship with Him.
jl: I appreciate you declaring your bias; the program in your footer returns 'god is superstition'. I will reciprocate by declaring my bias; I know God through my personal relationship with Him.

jc: To be real, this claim to a personal relationship with God would have to resolve to conversations with the Incarnation, for Jesus said no one comes to the Invisible God except through Him. All Christians make this claim, but fail with the smallest testing, in general the mind of rising men generating relationships by imagination, not by encounter.

Quote from: Student_t on June 08, 2013, 02:07:52 pm

This means that since the set of natural numbers is infinite, there cannot, in general, be a single greatest-possible being, because we can always conceive of a being with greater cognitive capabilities or knowledge (i.e, the ability to know n + 1 elements of the set).

jc: The trouble with this type of argument or thinking is that it requires minimal mental power or comprehension of mental power. All that is required is to think of three numbers, 1, 2, and 3, but the “extrapolation” from here to infinity is not very meaningful. Think of ants lugging grains of sand to their hill. One says, “You just take a lot of these grains, and soon you have a whole planet.” Though a brilliant ant, his thinking is really coming nowhere near the worldview of a usual man, encountering much larger amounts of soil and with a better idea how a planet is formed. (At least on the surface it is better.)

The ability to count to ten doesn’t offer a lot of leverage for comprehending God’s knowledge, or even human knowledge. For instance if you say to God, “The natural numbers,” you are thinking about your ten toes, but God is thinking about every atom in the cosmos, as well as the subatomic particles. There’s no bridge between these types of knowledge, though the one who can count his toes is sure God must be counting atoms like that, but He had a little more time. Instead there are huge regions of comprehending reality that are missing in the creatures with their limited power and limited perspectives.

jl: Regarding omniscience, your argument seems to have the following structure.
1. If God is omniscient, then God's knowledge is at least as great as the knowledge of any conceivable being.
2. If a being knows finitely many natural number, then there is a conceivable being whose knowledge of natural numbers is greater than the being’s.
3. If a being knows infinitely many natural numbers, then the being's knowledge is infinite.
4. Every being's knowledge is finite.
5. God is a being.
6. (By 5 and 4) God's knowledge is finite.
7. (By 6 and 3) God knows finitely many natural numbers.
8. (By 7 and 2) There is a conceivable being whose knowledge of natural numbers is greater than God's.
9. (By 8 and 1) God is not omniscient.

This is a good structure for an argument. However, the premises are presented without justification and are subject to critique.

Premise 1 (and a little of premises 2 and 3).
This premise implicitly assumes there is an "is at least as great as" relation between any being's knowledge and another's.

jc: This seems to argue my point, that can be stated various ways, such as God knows in depths but man knows in shallows, or that God knows spirit but man only knows matter. Yet the barrier isn’t just between God and man, but between man and the demigods or angels. You’d think the angelic superiority could’ve been included in religion, to inspire the humans. If a human says, “I’m able to conceive of a being whose knowledge is greater than an angel,” he fools himself, for his type of knowledge is incompatible with the knowledge of any angel. If you don’t know things in the same way, you cannot make remarks about the degree of knowledge the other possesses, only your own knowledge.

jl: First, there is no obvious (to me) reason that knowledge is subject to such a relation. This assumption thus requires justification.

jc: Importantly, it’s easy to demonstrate human knowledge to be externally pointed and selfishly motivated. And it’s easy to see angelic knowledge would face all realities inner and outer, and be selflessly motivated, even without angels here to give living examples. Swedenborg wrote about this often. The difference in knowledge means there can’t be authentic communication between humans and angels, except in a crude material sense.

jl: In addition, the narrative demonstrates this relation is the "is a superset of" relation. However, invoking the notion of greatness is an appeal to an external standard. As this standard is unexplored here, there is no justification for accepting that the standard greatness equates the "is at least as great as" relation on knowledge with the "is a superset of" relation on knowledge.

jc: Well-said. In this case the external standard for greatness appears to be merely holding more numbers in mind, upheld by those who can hold few numbers in mind. The argument is that men expect God to be like them, but able to count higher, and if He cannot count to infinity they’ll look for a greater God, “only absolutely omniscient Gods allowed.” One of my arguments has been that God does His best to forget men after they die since they are such violent and miserable sinners. Who’d ever want to remember that?

jl: It is worth noting that some hold these relations to be equivalent because of their beliefs about the greatness of knowledge.

jc: Yes, the human theory is that all are equal in their ideas, but it is a theory backed by guns so they already show their minds are of a low quality, not very distant from animals. Importantly, humans lack the ability to discuss the nature of knowledge, per se. The ability to gaze inwardly and examine the thinking process itself is beyond even the gurus. This means humans don’t even know what they’re saying, “All are equal in their ideas,” and in practice what they’re saying is, “Conform to our ideas or we’ll bring you trouble.”

jl: However, it is not clear that the language of sets and supersets applies to a being's knowledge at all.

jc: I’m only slowly understanding why you are bringing in a discussion of sets. A few examples would be helpful. I think you’re saying that humans who can count to ten on their toes have one set of knowledge, but they expect angels to count their toes and their fingers, which is a superset of knowledge. Yet they’re not admitting a fundamental difference in the very kernels of the ideas, as angels think selflessly, but they selfishly.

jl: It is conceivable that the limitations of set theory are too restrictive to adequately describe or apply to a being's knowledge.

jc: To push this inquiry farther we’d start asking the nature of the knowledge to understand or appreciate the natural numbers. Presumably this is based on actually counting things, but what is really going on in the mind as it counts, and what is it left with once the counting is through? To conceive of a greater number is always easy in some senses, for an angel or a human, you just add one digit to the left and you have ten times as many numbers represented. In this sense humans are unlimited, you just drag them out of bed each day and ask them to start adding numbers to the left on a lifelong list they are constructing, to prove their minds are greater than their neighbors’ who took a few days of vacation. Yet God’s knowledge, and angelic knowledge, if not human knowledge, is not limited to counting. Importantly, I’ve theorized the angels will be able to form rich, warm mental models for the personalities of thousands of friends, but we find humans only able to think in shallow ways even of relatives, that they don’t like. Humans count friends, but angels make friends that may be counted, knowing them well.

jl: Without the tools of set theory to guide the remainder of the original argument, premise one has no clear meaning and premise two is unjustified. It may or may not be true for whatever rules apply to relations between one being's knowledge and another's. Later, I will show this also brings premise three into question.

jc: You seem to be spotting a human limitation, that they conceive of thoughts as “things in the mind,” then whoever has the most things is greatest, ignoring the quality of ideas. The things in angelic minds are of a totally different nature than things in human minds, as Swedenborg attested. You could say the Venn diagram of the two has no intersection.

jl: Premise 2 (and a little of premise 4).
Consider the hypothetical that God knows all possible thoughts of all actual and conceivable beings. First, if there are infinitely many actual or conceivable beings who each have possible or actual knowledge of who they are (as distinct from each other being), then God's knowledge would be infinite. This presumes the rules of set theory apply to a being's knowledge (more on this later).

jc: You’re actually coming close to the mark of God’s knowledge, except He knows all possible thoughts of all actual beings, not all conceivable beings. Further, He can see the thoughts of His actual beings about any conceivable beings, are not worth much, especially when they also do not yet know themselves. Let them conceive of themselves first, as the angels do, then there is a chance of thinking realistically about other beings.

Since creatures trying to think about other beings matter little, the question is whether God can conceive of all possible thoughts of any beings of whom He can conceive. In this case “conceivable” is limited to what God chooses or is able to conceive of, since humans have virtually no useful imagination in this regard. (As the current example shows they can think about great counters, but not about great or glorious personalities.)

You’ve stumbled upon an interesting truth, that God limits His own infinity. If God has not conceived of it, no one ever will. I repeat humans offer no challenge in this regard, their ideas about differences trivial and foolish, such as “great counters.” Humans can conceive of unreal things, for instance a God who can create a planet in a finger-snap. All Christianity rests on such nonsense. That is not a useful contribution, it’s plain ignorance.

jl: Returning to the natural numbers, I can conceive of a being that knows the number one, and another being that knows the number two, and a third being that knows the number three, and so on. In this case, it might be convenient to say God would then know all natural numbers because He would know the thoughts of all these conceivable beings.

jc: This dances around an important truth, that human knowledge would be a subset of God’s knowledge, even their ideas about Him or attempts to prove His existence not really that important or interesting (from His perspective). Men like to think their conceptions bind God, instead He sees their limitations. But if your thoughts, jl, were known by God, then as you think about others He sees you thinking from within your potential. The question is whether you can think realistically about them, and are not making mere stick figures in your head of other greedy ones, as is usual human practice.

jl: However, I have not actually conceived of all these beings.

jc: Right, and if God does not conceive of them, they remain unconceivable, especially as God conceives of the soul with its powers, though humans conceive of minor math skills.

jl: Contrariwise, there may be natural numbers so large that it would take me longer than my entire life to conceive the number because of how large it is.

jc: MAY be? Surely you jest, weren’t we talking about infinite series? Of course there are, but more importantly how well are you conceiving of the numbers you believe you can conceive? For all humanity’s pride in its intelligence, I find no one even starting to conceive of the whole planet in its total time frame. The startling fact is that the world’s best science predicts an end to all major metals within a thousand years. Conceive that!

jl: Since I must actually conceive the number in order to conceive of a being that knows the number, such a being is inconceivable by me.

jc: Yes, and can you conceive of a thousand years? You’d be the first creature to do so. Since humans cannot look forward a thousand years, it is also revealed they are not conceiving well to look back a thousand years. Historians have a certain unreal quality.

jl: If there is any natural number so large that no being could conceive of it, then a being that knows this number is inconceivable.

jc: Again you seem lost on the basic concept of an infinite series. The series progresses, but what of nature? If the atoms in the universe are counted, why would God count higher? Again the point becomes trivial, since counting is not an advanced personal trait.

jl: In this case all natural numbers larger than this inconceivable number are also inconceivable; hence there are only finitely many conceivable natural numbers.

jc: Interestingly, the numbers don’t exist unless conceived. The series ends where God ceases to contemplate it. In theory it could be extended, but God chooses not to do so. You bring up another useful point, though, that writing down a number or thinking about a number doesn’t mean the number has really been conceived. It becomes a game of writing rather than of thought. To really conceive of it, it must be counted in some way. From observing humans, I’d say their ability to conceptualize large numbers ends with a football stadium. They do not think well about nations, or about billions using resources.

jl: Let us take this view for the moment (that there are finitely many conceivable natural numbers).

jc: I think the humans will oppose you on this, saying they can contemplate any number. To prove to them they are not conceptualizing well is not possible. They conceptualize as they can conceptualize, without an ability to think about one who can conceptualize better. This is an important reason why the churches look down on God, and why the Incarnations needed to speak in brief parables rather than writing the tomes that a race of sinners deserved and needed. To see another has superior traits is only feasible for angels.

jl: Restricted to natural numbers, this would only require that all of the finitely many conceivable natural numbers are known by God.

Jc. Not bad, but you mean of course the finitely many conceivable natural numbers that God can conceive, are known by God. This is infinitely beyond what creatures can know, since the distance between what they know and what God knows, is not conceivable by them. In practice I say a man can think of about 10,000 other humans as real, not more. This is why the cities have developed as they have, and problems of pollution are unseen. For a human to say it is real, it must fit in his senses. He can’t form broader conceptions.

jl: Thus, in this view, a being's knowledge of natural numbers can be both finite and greatest. This can be extended to all conceivable knowledge, not just conceivable knowledge about natural numbers. That is, the form of the argument applies to there being finitely many conceivable natural numbers and to there being finitely many conceivable elements of knowledge of any kind. Thus, in this view, a being's knowledge of everything can be both finite and greatest. In this case, premise two is false.

jc: Again, not bad. But God is Great, and His knowledge not conceivable by men. In fact the angels are great, their knowledge not conceivable by men. The distance from an angel to a human is greater than that from a human to an ant. This is a very intriguing argument you’re putting forth, that the infinities are practically limited by their conceivability. Yet to understand the limits of conception is another and far deeper project, on a world where men neither know or care the exact nature of a thought or how the mind stores it. Socrates gave a useful model, that the mind takes impressions like wax, but it requires power both to make and to hold the impression. Then you can also ask questions about the quality of the impressions, for instance noting angelic thoughts are always far superior to humans’.

jl: Now consider the view that there are no inconceivable natural numbers. Given our hypothetical (that God knows all possible thoughts of all actual and conceivable beings), we conclude that God knows all the natural numbers. In this case, premise four is false (slightly more on this premise later).

jc: I think it’s better to stick to proving humans have difficulty comprehending big numbers in any practical sense. This is why the planet is on the verge of self-destruction, each man only sees his own car and his own weekly trips to the landfill, utterly blind to the billions of others. Perhaps it is better to say infinity stops where God stops caring. God’s knowledge can thus be infinitely beyond human knowledge, in all practicality. Yet again, if the angels are bold they might prove their knowledge infinitely beyond, as well.

jl: All of this can be attacked by attacking the initial hypothetical, that God knows all possible thoughts of all actual and conceivable beings. If this is false, then the rest of my counterargument fails.

jc: You can amend my earlier suggestion, to that God knows all possible thoughts of all actual beings, and of all the beings of which He can conceive, where there are no challenges since humans are worthless at thinking about beings, blind to themselves too. Anyway as humans try to think of big numbers they are well within the limits God sets forth for the created souls, perhaps like an ant farm where the child who owns it has no worries those ants will lay a trap for him one day. To put it another way, as theologians try to go beyond God by thinking about big numbers, God sees it is just a little game, and they do poorly even with small numbers, unable to decide if a planet is overpopulated.

jl: It is worth noting, though, that some hold a version of this hypothetical as the definition of omniscience. For such people, my counterargument is a valid critique of the original argument.

jc: Actually, God can know all thoughts of actual and conceivable beings, without knowing all the natural numbers. If you want to be really strict, infinite series cannot be know by definition. No matter how many numbers there are you can always add one more, and this makes the entire question rather uninteresting. Even if God made the sole purpose of creation the contemplation of the full set of natural numbers, they still could not fit in because it is an endless set. So you were on the right track before. There is a limit to the infinities, where conception ends, either by choice or by necessity. That isn’t to say God hasn’t on some level counted the atoms in the universe, or that men would ever arise to understand how far above them this truly is. Once there are no more objects to count, God could starting theoretical objects, adding digits to the left of the sequence. A computer can do the same until it burns out, the limit of its “conceptions.”


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jayceeii

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Re: The Feasibility of Omniscience
« Reply #7 on: December 04, 2019, 02:52:48 pm »
Response to Jlouis, Part 2
@Student_t, I appreciate you declaring your bias; the program in your footer returns 'god is superstition'. I will reciprocate by declaring my bias; I know God through my personal relationship with Him.
jl: Premise 3.
Suppose the language of sets and supersets applies to a being's knowledge. Then the original argument holds that a being's knowledge is infinite if the cardinality of the being's knowledge is infinite. Premise three can then be split as follows. If a being knows infinitely many natural numbers, then a subset of the being's knowledge is infinite. If a subset of a being's knowledge is infinite, then the being's knowledge is infinite. In this case, the first part of the split follows from treating things a being knows as elements of the being's knowledge. That is, there is an injective function from the infinite set of natural numbers known by the being to the being's knowledge (treated as a set). Adding to this the formal rules of transfinite cardinality of sets, the cardinality of the being's knowledge is infinite. That is, we can conclude the second part of the split from the rules of set theory.

jc: I’m following most of this, but not agreeing with all of it perhaps because it isn’t clear to me what you are trying to say. It looked like you were saying if a being knows infinitely many natural numbers this is of course just a subset of his knowledge because he knows other things, but by itself enough to say his knowledge is infinite. In any case before explaining it, I’d ask you to explain why you think the natural numbers or other infinite series are not inherently inconceivable by the very definition under any mechanism of storing or comprehending. Conceiving of infinite series you are conceiving of what cannot be conceived, the conception you are forming only a rule, not the reality.

jl: Now suppose the language of sets and supersets does not apply to a being's knowledge. Then premise three cannot be split as above, because the split is not supported by the rules of set theory in this case. In particular, there is no rule to help us specify any conditions at all under which a being's knowledge would be infinite. Could it be that a being knowing the set of all natural numbers is consistent with the being's knowledge being finite? There is no rule that produces a contradiction with this. Perhaps knowing the whole set only constitutes one element of knowledge.

jc: Again we are not quite connecting over basics, and I’d return to asking questions about the degree and quality of the knowledge. For instance humans thinking about their neighbors have generally in mind, “Some grabby guys who amount to competitors.” They don’t form deep friendships, because they don’t see much when they look at others or at themselves. If you think all the natural numbers can be known, you aren’t understanding the natural numbers, as I was seeing some evidence above that you were surprised there would be numbers too big to be conceived. Perhaps you’ll explain yourself more clearly.

Look at it this way, an infinite series is by definition not a set. You can look at portions of it and call these sets, but the whole never ends even when the universe runs out of countable quarks. Yet for the humans the series ends when they go to a football game. This is their actual idea about “the whole world,” not seven billion ruining the future. I suppose what I am saying is that God cannot be infinite according to rules, particularly those that humans can devise. A rule is a powerful thing, but humans don’t follow the rule a long ways into comprehending what the rule generates. Perhaps angels do better.

jl: This may sound strange, perhaps even absurd. However, the field of mathematics is replete with infinite sets specified all at once with one definition. The set of all prime numbers is the set of natural numbers that have exactly two divisors. The set of natural numbers is the set of entities that result from repeated applications of the successor function (using Peano's Axioms to formalize the set of natural numbers). The set of rational numbers is the set of ratios of the form a / b where a and b are integers and b is not zero.

jc: Indeed, the natural numbers are a sparse infinity, where the set of rational numbers is a dense infinity, but perhaps you see these can only be called sets in the loosest way.

jl: Do these sets defined by properties constitute three elements of knowledge or infinitely many elements of knowledge?

jc: Will you admit these are not true or complete sets, and inherently cannot be known? Theology in general has been holding God to absurd standards, but this does not take away from God’s actual grandeur, power or knowledge. They’re looking in the wrong place. Men are seeking an infinite omniscient God, while hating the neighbor and denuding a planet. Another thing God can’t do is make a world in a jiffy; just wait for it.

jl: Of course, this approach can be scrutinized; and, it should be scrutinized. However, this can serve as a starting point for exploring the validity of premise three. It is unclear that the rules of set theory apply or ought to apply to a being's knowledge. This is unjustified in the original argument. If the rules do not apply, is there a coherent model of knowledge that is inconsistent with premise three? I would like to explore this more with a willing participant.

jc: I liked where you were going before, that the infinities are practically limited, when there’s no one to conceive them. For God to count more than the atoms in the universe would mean counting the dots in His Mind, and are creatures any more than these dots?

jl: Premise 4.
This premise is presented in the narrative as being obviously true. However, there is no justification presented for it. It is logically independent of the other premises; that is, it does not follow from the other premises. And, I do not believe is it true. That is, I have no philosophical issue with the negation of premise four. Without justification of this premise, the original argument is unconvincing.

jc: I’d again assert God’s knowledge to be infinitely beyond the creatures, including angels, but that “infinite” is defined by incomprehensibility, not by rules that make them feel they are standing right beside Him. A true infinity by definition cannot be comprehended, even were God to make the entire creation about comprehending this.

jl: Premise 5.
I have no reason to doubt this premise. It follows directly from the definition of God.

jc: God is the Creator, and is certainly not a “being” in the same sense as the created souls or creatures are beings. The Doctrine of the Trinity is on hand to begin explaining this, three living aspects of one God, where creatures manifest one, or just part of one. Thinking about God a certain reverence is required, and the wise when they meet the Incarnation recognize they are not talking with a being like themselves. This is the real secret of why the disciples responded to Jesus, not because He asked them to cast nets.

jl: Conclusion.
The OPs argument was described using the language of formal logic. Various reasons are given to doubt the applicability or validity of some premises of this argument. Other premises are simply unsupported. The result is that the original argument is unconvincing. I welcome critique of my response so long as it is made with a good faith effort to reach a common understanding.

jc: Ha, you have admitted the knowledge of individuals is not necessarily compatible, so who’s to say my knowledge is compatible with yours, that we can reach a “common understanding”? Perhaps even after we’d both agree to certain word combinations, my idea of what these things mean might not correspond with yours. But perhaps we could have fun with it, not a “debate to the death” as it is among the humans. Looking again at the original argument, if you accept that the infinities inherently cannot be known then this is a wrong definition of “omniscient,” if we want to go on using that word. Omniscient can also mean all that can be known, or that God chooses to know, but in any case God surrounds the lives of all creatures by His omniscience though they don’t see it.