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This question came up here recently (see later half of my post#9):

https://www.physicsforums.com/threads/how-can-we-construct-ordinals-after-large-veblen.934141/

Possibly post#4 may be of some relevance too.

I have already described my view more or less at length before (post#83 and post#169) in the following rather extensive (old) thread:

https://www.physicsforums.com/threads/penrose-chess-problem.911538

So there is no need of repeating it (apart from if there is a need to clear-up some point in detail) .

Since this question came up again (though somewhat indirectly) in this recent thread, I supposed it would be interesting to gather various opinions on this (the question is of mostly mathematical interest than practical one). And this is the purpose (so if you partially/totally disagree with me, feel free to express so).

@Deedlit Sorry for tagging so many times (didn't know there would be many threads all of a sudden), what is your opinion/understanding of this? Or of any other experts (on this specific topic) that read this?

Below I will just quote what number of (eminent) people have said regarding this (with a brief sentence or two in the beginning adding some context) ... well possibly to generate some interest :p. If some quotes are slightly (or more than slightly) out-of-context, then my apology. Just as one example, opinions can change sometimes.

(i) Kurt Godel

"Kurt Godel Collected Works Volume II -- Some remarks on the undecidability results (1972a)"

What I have read is that it isn't clear whether this note was meant for publication.

(ii) Stephen Kleene

This is a comment on remarks in (i).

Reflections on Church's thesis

(iii) Solomon Feferman

Even though I have quoted the full sentence only for the sake of completion, it is the part that I have underlined that I believe is important here. Partly because he wasn't specifically talking about number-theoretic functions, but a more generic point. And given some of the most important work of the author seems to be quite related to the topic at hand, it seems fair enough to quote this.

https://math.stanford.edu/~feferman/papers/penrose.pdf

Penrose's Godelian Argument

I have removed the post-script and some quotes to keep the OP relatively short (and mainly just relevant to actual question).

https://www.physicsforums.com/threads/how-can-we-construct-ordinals-after-large-veblen.934141/

Possibly post#4 may be of some relevance too.

I have already described my view more or less at length before (post#83 and post#169) in the following rather extensive (old) thread:

https://www.physicsforums.com/threads/penrose-chess-problem.911538

So there is no need of repeating it (apart from if there is a need to clear-up some point in detail) .

Since this question came up again (though somewhat indirectly) in this recent thread, I supposed it would be interesting to gather various opinions on this (the question is of mostly mathematical interest than practical one). And this is the purpose (so if you partially/totally disagree with me, feel free to express so).

@Deedlit Sorry for tagging so many times (didn't know there would be many threads all of a sudden), what is your opinion/understanding of this? Or of any other experts (on this specific topic) that read this?

Below I will just quote what number of (eminent) people have said regarding this (with a brief sentence or two in the beginning adding some context) ... well possibly to generate some interest :p. If some quotes are slightly (or more than slightly) out-of-context, then my apology. Just as one example, opinions can change sometimes.

(i) Kurt Godel

"Kurt Godel Collected Works Volume II -- Some remarks on the undecidability results (1972a)"

What I have read is that it isn't clear whether this note was meant for publication.

A philosophical error in Turing's work

Turing in his 1937, page250 (1965, page 136), gives an argument which is supposed to show that mental procedures cannot go beyond mechanical procedures. However, this argument is inconclusive. What Turing disregards completely is the fact that mind, in its use, is not static, but constantly developing, i.e., that we understand abstract terms more and more precisely as we go on using them, and that more and more abstract terms enter the sphere of our understanding. There may exist systematic methods of actualizing this development, which could form part of the procedure. Therefore, although at each stage the number and precision of the abstract terms at our disposal may be finite, both (and, therefore, also Turing's number of distinguishable states of mind) may converge toward infinity in the course of the application of the procedure. Note that something like this indeed seems to happen in the process of forming stronger and stronger axioms of infinity in set theory. This process, however, today is far from being sufficiently understood to form a well-defined procedure. It must be admitted that the construction of a well-defined procedure which could actually be carried out (and would yield a non-recursive number-theoretic function) would require a substantial advance in our understanding of the basic concepts of mathematics. Another example illustrating the situation is the process of systematically constructing, by their distinguished sequences ##\alpha_n \rightarrow \alpha##, all recursive ordinals ##\alpha## of the second number-class.

(ii) Stephen Kleene

This is a comment on remarks in (i).

Reflections on Church's thesis

Let me now address an argument reported in [29], p. 222 (after Wang [28],

p. 325):"Godel . . . objects that Turing 'completely disregards' that:

(G) 'Mind, in its use, is not static, but constantly developing.'

. . . Gόdel granted that

(F) The human computer is capable of only finitely many internal (mental)

states.

holds 'at each stage of the mind's development', but says that

(G)' '. . . there is no reason why this number [of mental states] should not

converge to infinity in the course of its development.'"

If one chooses to believe (G)', I can see that it would imply that there need be no end to the possibilities for the human mind to invent stronger and stronger and stronger . . . formal systems that would, in the face of Godel's ever renewing incompleteness theorem, decide more and more and more . . . number theoretic

propositions ...

But I reject that (G)' could have any bearing on what number-theoretic functions are effectively calculable.

(iii) Solomon Feferman

Even though I have quoted the full sentence only for the sake of completion, it is the part that I have underlined that I believe is important here. Partly because he wasn't specifically talking about number-theoretic functions, but a more generic point. And given some of the most important work of the author seems to be quite related to the topic at hand, it seems fair enough to quote this.

https://math.stanford.edu/~feferman/papers/penrose.pdf

Penrose's Godelian Argument

I must say that even though I think Godel’s incompleteness theorems are among the most important of modern mathematical logic and raise fundamental questions about the nature of mathematical thought, and even thoughI am personally convinced of the extreme implausibility of a computational model of the mind, Penrose’s Godelian argument does nothing for me personally to bolster that point of view, and I suspect the same will be true in general of similarly inclined readers.

**Edit:**I have removed the post-script and some quotes to keep the OP relatively short (and mainly just relevant to actual question).

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