Gödel’s Infinite Candy Store

[Eugene I]

I'm using "computability" as an analogy for "experience".

But it is ironic to me that you are using abstract mathematics to deny existence to the experience of what you call the "abstract" idea of infinity wholeness.

OK, here is a simple question: we know that the abstract idea of matter exists. Then why idealism claims that matter itself does not exist? That is because if an abstract idea of something exists, it does not necessarily mean that the "something" that this idea is about also actually exists and can be experienced.

But I can see the contradiction that you are pointing to (using the number analogy again): if every single possible number exists then how come the infinity of all numbers does not? There are two possible answers:
1. Platonic answer. Yes, every possible number actually exists. But you are right, in this case the whole infinity of all numbers necessarily has to exist.
2. Non-Platonic answer: any number actually exists only at the instance when it is actually computed/experienced. In this case every possible number does not always exist (until it is computed), and therefore the actual infinity of all numbers does not actually exist as well.

But as I said above, if we extrapolate the statement #1 to the set of all possible ideas, we run into the Russel's paradox (because the set of all ideas is itself an idea and therefore must include itself), and that is a very serious one. The only way around it that mathematicians found so far is just to exclude the objects like "the set of all sets that include itself" from existence in mathematics (which was accomplished in the Zermelo-Fraenkel set theory). And the same logic would equally apply to the set of all ideas that includes itself.

But apart from the the Russel paradox, the specific statements (#1 or #2) both seem to be logically consistent. And as I said, to me the question of which one it true is undecideable because I have no way to prove or disprove ether one. Among mathematicians there are as many proponents of #1 as there are of #2. But as far as I know, the fact is that nobody so far was able to experience the whole infinity of all numbers (as well as the whole infinity of all ideations), so there is no experiential evidence that such thing actually exists.

[AshvinP]

I'm using "computability" as an analogy for "experience".

But it is ironic to me that you are using abstract mathematics to deny existence to the experience of what you call the "abstract" idea of infinity wholeness.

OK, here is a simple question: we know that the abstract idea of matter exists. Then why idealism claims that matter itself does not exist? That is because if an abstract idea of something exists, it does not necessarily mean that the "something" that this idea is about also actually exists and can be experienced.

You were not only using it as an analogy, though, but claiming what is incomputable according to mathematical arguments is also incapable of being experienced. Maybe that's not what you intended. We don't need to argue that - suffice to say, I disagree.

Two things here - 1) I am not claiming "wholeness of infinity" exists simply because I can abstractly conceive of it (rather I think it may flow from your conclusion re: Godel theorem), 2) in a meaningful sense (the only sense there is under idealism), "matter" does exist - it is what we call images in the sense-world that are devoid of Spirit from our 1st-person perspective (the only perspective there is). The question of why they are devoid of Spirit in the modern age is a massive one and I am exploring that in the integral myth essays.

But I can see the contradiction that you are pointing to (using the number analogy again): if every single possible number exists then how come the infinity of all numbers does not? There are two possible answers:
1. Platonic answer. Yes, every possible number actually exists. But you are right, in this case the whole infinity of all numbers necessarily has to exist.
2. Non-Platonic answer: any number actually exists only at the instance when it is actually computed/experienced. In this case every possible number does not always exist (until it is computed), and therefore the actual infinity of all numbers does not actually exist as well.

To date, Cleric still has the best illustration of why #2 is pragmatically equivalent to #1, which I frequently quote but will quote again (my emphasis):

What I said above answers this question. It's useless to try to imagine pure ideas without experience, precisely because, as you say, we only create a hard problem for ourselves. Yet this doesn't preclude the fact that the experienced ideas exist in certain relations. To give a simplified example, if I think about 1 and 2, then 4 and 5, does this mean that 3 doesn't exist until it is experienced? From experiential perspective every idea exists for me only when I experience it. But still, the relation between 2 and 4 is such that they can only be what they are if there's 3 in between. That's why I've always said (when you bring the Platonism argument) that it's irrelevant to me to fantasize some abstract container for ideas, which I can never experience in its purity. The important thing is that when I discover 3, nothing really changes for 1,2,4,5 - they are only complemented, the ideal picture becomes more complete. Even if 3 was never discovered, the relation between the above numbers would be as if 3 exists. This would be different if after the discovery of 3 all other numbers change relations. Then we would really have justification to speak of ideas being created. The act of creation of the idea has measurable effect and displaces all other ideas in some way. But as long as I discover ideas and beings, which only complement my own experiential ideal landscape, all talks about if these ideas and beings exist in 'pure form' before I experience them, is pointless

But as I said above, if we extrapolate the statement #1 to the set of all possible ideas, we run into the Russel's paradox (because the set of all ideas is itself an idea and therefore must include itself), and that is a very serious one. The only way around it that mathematicians found so far is just to exclude the objects like "the set of all sets that include itself" from existence in mathematics (which was accomplished in the Zermelo-Fraenkel set theory). And the same logic would equally apply to the set of all ideas that includes itself.

But apart from the the Russel paradox, the specific statements (#1 or #2) both seem to be logically consistent. And as I said, to me the question of which one it true is undecideable because I have no way to prove or disprove ether one. Among mathematicians there are as many proponents of #1 as there are of #2. But as far as I know, the fact is that nobody so far was able to experience the whole infinity of all numbers (as well as the whole infinity of all ideations), so there is no experiential evidence that such thing actually exists.

In my view, your use of Russel's paradox here is basically saying, "the abstract intellect cannot conceive of whole infinity, therefore it does not exist." That is very similar to Schopenhauer's claim that, since mythological symbols are rife with "paradoxes" when viewed by the abstract intellect, it must not be pointing to any deeper and coherent spiritual reality. That simply ignores the possibility that abstract intellect is not the fundamental max capacity of perception-cognition.

The question of whether anyone has experienced or come closer to experiencing whole infinity or eternity is a more complicated one. Philosophically speaking, if we accept the mumorphic formulation, then we must be experiencing that aspect all the time. Also from the phenomenological approach, which is what Cleric's quote above is referencing, we can also conclude that whole infinity pragmatically exists. But even if you do not accept either of those, I think your own conclusion re: Gödel's theorem also necessitates the pragmatic existence of whole infinity in order to remain internally logically coherent.

[Eugene I]

In my view, your use of Russel's paradox here is basically saying, "the abstract intellect cannot conceive of whole infinity, therefore it does not exist."

No-no.

1. I'm not saying that if "the abstract intellect cannot conceive of whole infinity, therefore it does not exist." I'm only saying that there is no experiential proof (so far) that the whole infinity exists simply because nobody has ever actually experienced it. That does not mean that it does not exist, it simply means what it means: its existence is not yet experimentally proven. That is why I sad that I (at this moment) have no way to prove of disprove the statement the "whole infinity exists", so it is undecidable for me.

2. The Russel's paradox has nothing to do with the fact that "the abstract intellect cannot conceive of whole infinity", it has to do with the fact that the infinite set of all ideas is itself an idea, therefore it must include itself, but this is not possible because, as Russel proved, it leads to logical contradiction.

[AshvinP]

In my view, your use of Russel's paradox here is basically saying, "the abstract intellect cannot conceive of whole infinity, therefore it does not exist."

No-no.

1. I'm not saying that if "the abstract intellect cannot conceive of whole infinity, therefore it does not exist." I'm only saying that there is no experiential proof (so far) that the whole infinity exists simply because nobody has ever actually experienced it. That does not mean that it does not exist, it simply means what it means: its existence is not yet experimentally proven. That is why I sad that I (at this moment) have no way to prove of disprove the statement the "whole infinity exists", so it is undecidable for me.

2. The Russel's paradox has nothing to do with the fact that "the abstract intellect cannot conceive of whole infinity", it has to do with the fact that the infinite set of all ideas is itself an idea, therefore it must include itself, but this is not possible because, as Russel proved, it leads to logical contradiction.

Eugene - you are equating Russel's mathematical argument, which is beyond any reasonable doubt an abstract intellectual model, with "experiential proof". Once again, this was the main point of Cleric's post - we cannot assume that what has not been captured by abstract intellectual models (including mathematical ones) has not been experienced, let alone cannot be proven to exist. To use my much more crude analogies, that is like a 2-D being of width-height saying 3-D reality of width-height-depth has not been experienced and cannot be proven to exist because it cannot be captured by his own 2-D terms of width-height.

[Eugene I]

Eugene - you are equating Russel's mathematical argument, which is beyond any reasonable doubt an abstract intellectual model, with "experiential proof". Once again, this was the main point of Cleric's post - we cannot assume that what has not been captured by abstract intellectual models (including mathematical ones) has not been experienced, let alone cannot be proven to exist. To use my much more crude analogies, that is like a 2-D being of width-height saying 3-D reality of width-height-depth has not been experienced and cannot be proven to exist because it cannot be captured by his own 2-D terms of width-height.

I'm repeating again that the Russel's argument has nothing to do with the "experiential proof". The Russel's argument is that the idea of all sets of possible ideas is logically self-contradicting.

I agree with your argument: the absence of the experimental proof that "X exists" is not a proof that "X does not exist", so there is still a possibility the "X exists" even when there is no experiential evidence for it. Obviously it is not possible to prove the non-existence of anything.

So:
1. There is no experiential evidence that the set of all ideas actually exists.
2. The #1 above does not mean, and is not a proof, that it does not exist, so there is still a possibility that it does exist.
3. However, the existence of the set of all ideas is logically self-contradicting (according to the Russel's paradox). You may still argue that the Aristotelian logic does not apply to the set of all ideas (I'm making it easier for you )

[AshvinP]

Eugene - you are equating Russel's mathematical argument, which is beyond any reasonable doubt an abstract intellectual model, with "experiential proof". Once again, this was the main point of Cleric's post - we cannot assume that what has not been captured by abstract intellectual models (including mathematical ones) has not been experienced, let alone cannot be proven to exist. To use my much more crude analogies, that is like a 2-D being of width-height saying 3-D reality of width-height-depth has not been experienced and cannot be proven to exist because it cannot be captured by his own 2-D terms of width-height.

I'm repeating again that the Russel's argument has nothing to do with the "experiential proof". The Russel's argument is that the idea of all sets of possible ideas is logically self-contradicting.

I agree with your argument: the absence of the experimental proof that "X exists" is not a proof that "X does not exist", so there is still a possibility the "X exists" even when there is no experiential evidence for it. Obviously it is not possible to prove the non-existence of anything.

So:
1. There is no experiential evidence that the set of all ideas actually exists.
2. The #1 above does not mean, and is not a proof, that it does not exist, so there is still a possibility that it does exist.
3. However, the existence of the set of all ideas is logically self-contradicting (according to the Russel's paradox). You may still argue that the Aristotelian logic does not apply to the set of all ideas (I'm making it easier for you )

If Russel's argument has nothing to do with experiential proof, as you stated in the bolded assertion (we may have conflated "experiential" and "experimental", but I don't think that matters), then why do you keep making the underlined assertions as if they are conclusive facts? If those assertions are not based on Russel's argument, then what are you basing them on?

Also, you skipped over the pragmatic argument that #2 is equivalent to #1 (what you call "Platonic" argument), which indicates that we can and do have experiential proof of whole infinity (given the conclusion from Godel's theorem).

[Eugene I]

2) in a meaningful sense (the only sense there is under idealism), "matter" does exist - it is what we call images in the sense-world that are devoid of Spirit from our 1st-person perspective (the only perspective there is). The question of why they are devoid of Spirit in the modern age is a massive one and I am exploring that in the integral myth essays.

No, I was talking about the existence in a different sense - in the "actual"/"existential" sense. Materialists claim that matter not only exists as images in the sense-world, but it actually fully exists as reality apart from consciousness. And they actually hold this idea in their individual minds that " matter actually exists as reality apart from consciousness". Now, they indeed hold this idea and consciously experience it from their 1-st person perspective. Yet this fact that they hold this idea does not mean that matter actually exists as reality apart from consciousness.

So, again, this is an illustration that the fact that an idea about "something" ("X") exists and that this idea can be held and experienced in consciousness does not mean that the "X" itself (of which this idea is about) necessarily also exists as an actual reality. This statement can be applied to the idea of the set of all ideas: you can have an idea of the set of all ideas, but that does not necessarily mean that such set by itself actually exists and can be actually fully experienced. Similarly, everyone can have an abstract idea of the infinity of natural numbers, but nobody was able to actually compute and experience the whole infinity of all natural numbers. But again, that does not mean and does not prove that such infinity does not exist.

[AshvinP]

2) in a meaningful sense (the only sense there is under idealism), "matter" does exist - it is what we call images in the sense-world that are devoid of Spirit from our 1st-person perspective (the only perspective there is). The question of why they are devoid of Spirit in the modern age is a massive one and I am exploring that in the integral myth essays.

No, I was talking about the existence in a different sense - in the "actual"/"existential" sense. Materialists claim that matter not only exists as images in the sense-world, but it actually fully exists as reality apart from consciousness. And they actually hold this idea in their individual minds that " matter actually exists as reality apart from consciousness". Now, they indeed hold this idea and consciously experience it from their 1-st person perspective. Yet this fact that they hold this idea does not mean that matter actually exists as reality apart from consciousness.

So, again, this is an illustration that the fact that an idea about "something" ("X") exists and that this idea can be held and experienced in consciousness does not mean that the "X" itself (of which this idea is about) necessarily also exists as an actual reality. This statement can be applied to the idea of the set of all ideas: you can have an idea of the set of all ideas, but that does not necessarily mean that such set by itself actually exists and can be actually fully experienced. Similarly, everyone can have an abstract idea of the infinity of natural numbers, but nobody was able to actually compute and experience the whole infinity of all natural numbers. But again, that does not mean and does not prove that such infinity does not exist.

I know what you are saying here, and it is actually a much deeper topic, one that is not at all relevant to our other arguments, because I never claimed the abstract idea of something means the ideal content it is referring to exists. Even if that is true, I am not making that argument for "whole infinity". Rather I am making the arguments already outlined above. To which I will add:

"You may still argue that the Aristotelian logic does not apply to the set of all ideas" - that is a part of what I have been arguing re: mumorphic formulation. If we accept that formulation, then we need to also admit the "polar logic" which applies to the spiritual realm, and not only the Aristotelian logic which applies to the sense-world.

I am going to start a new thread about this question of "does matter actually exist".

[Eugene I]

Eugene - you are equating Russel's mathematical argument, which is beyond any reasonable doubt an abstract intellectual model, with "experiential proof". Once again, this was the main point of Cleric's post - we cannot assume that what has not been captured by abstract intellectual models (including mathematical ones) has not been experienced, let alone cannot be proven to exist. To use my much more crude analogies, that is like a 2-D being of width-height saying 3-D reality of width-height-depth has not been experienced and cannot be proven to exist because it cannot be captured by his own 2-D terms of width-height.

I'm repeating again that the Russel's argument has nothing to do with the "experiential proof". The Russel's argument is that the idea of all sets of possible ideas is logically self-contradicting.

I agree with your argument: the absence of the experimental proof that "X exists" is not a proof that "X does not exist", so there is still a possibility the "X exists" even when there is no experiential evidence for it. Obviously it is not possible to prove the non-existence of anything.

So:
1. There is no experiential evidence that the set of all ideas actually exists.
2. The #1 above does not mean, and is not a proof, that it does not exist, so there is still a possibility that it does exist.
3. However, the existence of the set of all ideas is logically self-contradicting (according to the Russel's paradox). You may still argue that the Aristotelian logic does not apply to the set of all ideas (I'm making it easier for you )

If Russel's argument has nothing to do with experiential proof, as you stated in the bolded assertion (we may have conflated "experiential" and "experimental", but I don't think that matters), then why do you keep making the underlined assertions as if they are conclusive facts? If those assertions are not based on Russel's argument, then what are you basing them on?

I'm basing them on a simple fact that myself (having mathematical background) I have never experienced neither the actual infinity of all natural numbers, nor the actual infinity of all possible ideas. And I have never heard of anyone claiming that he/she actually experienced those. That's all I'm saying.

Also, you skipped over the pragmatic argument that #2 is equivalent to #1 (what you call "Platonic" argument), which indicates that we can and do have experiential proof of whole infinity (given the conclusion from Godel's theorem).

The existence of the set of all mathematical statements does not follow from Godel theorem at all, I don't know how you would arrive to such conclusion. Godel's theorem states that any finite set of mathematical axioms (that includes arithmetics) is always incomplete and contains undecidable statements. Now, each of these undecidable statements can be further added to that set of axioms to create a new theory with the extended set of axioms. So, this theorem describes an iterative procedure of adding axioms one-by-one to the previously formulated theories, just like the algorithm x(n+1)=x(n)+1 describes a procedure of adding ones to the already pre-computed natural numbers. But the statement that the infinite set of all mathematical statements (and all theories) actually exists does not follow from the Godel's theorem at all, just like the statement that the infinity of all natural numbers actually exists does not follow from the iterative algorithm x(n+1)=x(n)+1.

[AshvinP]

I'm repeating again that the Russel's argument has nothing to do with the "experiential proof". The Russel's argument is that the idea of all sets of possible ideas is logically self-contradicting.

I agree with your argument: the absence of the experimental proof that "X exists" is not a proof that "X does not exist", so there is still a possibility the "X exists" even when there is no experiential evidence for it. Obviously it is not possible to prove the non-existence of anything.

So:
1. There is no experiential evidence that the set of all ideas actually exists.
2. The #1 above does not mean, and is not a proof, that it does not exist, so there is still a possibility that it does exist.
3. However, the existence of the set of all ideas is logically self-contradicting (according to the Russel's paradox). You may still argue that the Aristotelian logic does not apply to the set of all ideas (I'm making it easier for you )

If Russel's argument has nothing to do with experiential proof, as you stated in the bolded assertion (we may have conflated "experiential" and "experimental", but I don't think that matters), then why do you keep making the underlined assertions as if they are conclusive facts? If those assertions are not based on Russel's argument, then what are you basing them on?

I'm basing them on a simple fact that myself (having mathematical background) I have never experienced neither the actual infinity of all natural numbers, nor the actual infinity of all possible ideas. And I have never heard of anyone claiming that he/she actually experienced those. That's all I'm saying.

You cannot say that unless you also claim that you have raised your entire subconscious experience into the light of conscious awareness. There are many (infinite) experiences we are having now that we are not aware of. Under my idealist view, claiming to raise entire subconscious into consciousness would be quite bold, because it would mean we are claiming to have attained to full status of the Spirit, the One, God, etc.. And I have heard of people claiming they have experienced God, Oneness, etc., as well as people who claim they have experienced vastly expanded cognition which cannot reasonably doubt the existence of eternity or whole infinity.

Also, you skipped over the pragmatic argument that #2 is equivalent to #1 (what you call "Platonic" argument), which indicates that we can and do have experiential proof of whole infinity (given the conclusion from Godel's theorem).

The existence of the set of all mathematical statements does not follow from Godel theorem at all, I don't know how you would arrive to such conclusion. Godel's theorem states that any finite set of mathematical axioms (that includes arithmetics) is always incomplete and contains undecidable statements. Now, each of these undecidable statements can be further added to that set of axioms to create a new theory with the extended set of axioms. So, this theorem describes an iterative procedure of adding axioms one-by-one to the previously formulated theories, just like the algorithm x(n+1)=x(n)+1 describes a procedure of adding ones to the already pre-computed natural numbers. But the statement that the infinite set of all mathematical statements (and all theories) actually exists does not follow from the Godel's theorem at all, just like the statement that the infinity of all natural numbers actually exists does not follow from the iterative algorithm x(n+1)=x(n)+1.

Honestly, I don't know if it follows from Godel's theorem or not. But it does follow from your previous formulation of what you also wrote above:

I gave you a simple illustration from mathematics: a never-ending iterative algorithm of adding +1 actually exists and you can compute and experience every possible iteration of it. According to this algorithm, the sequence of natural numbers is inexhaustible and the execution of the iterative algorithm is never ending (because you cannot reach a number x(n) to which you can not add 1 and compute a larger number x(n)+1)

If we consider each natural number an ideal form, which I think we both agree is reasonable to do, and it is what we have been doing, then the fact that they will always exist in definite relation to each other, whether we experience them or not, means the infinity set of ideal forms actually exists under the pragmatic-phenomenological approach. As soon as we discover an ideal form, it is as if that ideal form always existed (unless you claim discovering them will retroactively change all of the previous relations).