Re: Am I a crank?

MoeBlee wrote:
Tony Orlow wrote:
Hi Aatu -
You say here that induction follows from our mathematical picture of the
naturals,

He's talking about a particular axiom of induction in PA. I agree that
it essential to our understanding of the naturals. But you should
understand that induction is even more general. Inductive sets are a
certain kind of set. Roughly, an inductive set is a set that is the
intersection of a class of sets each of which is closed under a given
relation.

Yes. But it seems our understanding of such inductively generated sets is much more basic than our understanding of any set theoretic ideas. We readily understand natural numbers, various formal expressions in formal languages, and so forth, without any recourse to set theoretic thinking of these things as the smallest set closed under this or that rule. Such set theoretic reasoning and understanding only becomes important when we reach inductive definitions - such as arithmetical truth or being an ordinal notation in O - of more generalized nature. Taking these things to fall under the more generalized notion of inductive definition is mathematically fruitful, but does not reflect anything of conceptual or "foundational" significance.

I don't speak for Koskensilta, but this may work in many directions -
first order PA may be thought of as a formalization of our mathematical
understanding, so that it is not required to reverse this as you are
doing.

In some sense we can take PA as a "formalization of our mathematical understanding" - in that it does indeed incorporate a part of our mathematical knowledge, or beliefs - and in others such a identification would be horribly wrong. What is central to our understanding of the naturals is that they are generated from a single element, 0, by repeated application of the successor operation. As said, abstracting on this we get the informal induction principle, applying to any property we regard as determinate. In particular, if we accept the language of arithmetic as meaningful, induction will apply to properties definable in that language, giving us PA. Of course, if we accept the language of arithmetic as meaningful, we should accept as meaningful also properties definable by quantification over properties thus definable, and so forth. There is no formal theory we could recognize, only on basis of our acceptance of the induction principle and the intelligibility of the language of arithmetic, as correct, that would cover all principles that are acceptable on such basis, for Gödelian reasons. Pondering on such things, we recognize that such is the case even in case of set theory, and indeed any basis we can come up for mathematics. Formalized theories can only capture a portion - perhaps the only interesting portion! - of what our informal ideas lead us to accept, or what is acceptable on basis of our informal ideas.

Generally, I don't see existence of naturals as arrived upon inductively. In first order PA, the natural numbers are not mentioned in the theory itself.

True indeed, but first order PA is entirely irrelevant to our understanding of natural numbers.

We do have transfinite induction in Z set theories. It just does not
prove what you want it to prove here. So get some axioms that prove
what you want them to prove.

A silly request. People like Orlow are not interested in mathematical proofs or derivations as usually understood, but in heroic resistance to the evil establishment.

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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