Re: Choice sequences, intuition, etc

Here is part 2 as promised.

Keith Ramsay <kram...@xxxxxxx> wrote:

see offhand how to avoid. But this is in fact one of the
cases where the use of the law of excluded middle can be
removed automatically.

Yes, I had a feeling that there was a meta-theorem that so applied,
in one-quantifier cases. It might be different if someone proved
something on the basis that TwinPrime was true, and also on false.

I hope you're not identifying "meaningful" with "is either
true or false".

Heavens no! For arithmetical statements, I take meanings
as usual, in the Tarski/Hilbert sense, i.e. inductively
according to the orthodox meaning of quantifiers.
It can be re-worded in other ways.

OC meaning for Analysis statements might be somewhat
more dubious, and for Set theoretic ones, wildly so.

Imagine that you've learned the statement of the continuum
hypothesis... ...Now someone shows you the proof of some
result... ...assuming the continuum hypothesis.

If the statement was an arithmetic one, then I would regard it as
almost untouched (alethically) by the proof; ditto for not-CH.

However, many such proofs (e.g. Friedman's) rely only
on the *consistency* of the assumption, not its truth.
Here, the evidence is very much stronger.

All the moves which try to treat being true as necessarily
entailing anything at all about the possibility of being
verified in principle, however one might imagine it, are
treated as "anti-realist".

In this general area, it seems to me there is a big difference
between science and math. In the former, such positivist
leanings have some force; in the latter, not, IMHO.

|Well, for the purposes of this thread, this truth v proof
|matter is not my "concern".

You wrote: [etc]

You are quite right - once the debate was under way,
it naturally veered onto such things. But my original concern,
for which I started the thread, was to clarify just what
the term "free choice sequence" actually *meant*.

With Herman Jurjus' help, I seem to be getting some idea.
I hope to get it better still.

| So again - Where is the Halmos of intuitionism?

Heh, I find this idea amusing.

Yes, it's rather fun, innit! :)

-- 2nd reading of Bill of writes.

.

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