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*Post by m***@wp.pl**Post by David Bernier**Post by m***@wp.pl**Post by David Bernier**Post by m***@wp.pl**Post by David Bernier**Post by m***@wp.pl*What Goedel did was mechanize the reasoning that

leads us to say "We don't know everything". He did

that by representing our reasoning with numbers

and arithmetic expressions, by fully mechanizing it.

Godel's formalism states it clearly - there is

no implication between (a is proven) and a. Right?

Tell me, do You often say "we've proven that all

consistent theories [of some kind] are incomplete,

so we can't know whether they are or not"

?

Somehow I've never meet a mathematician talking

this way. Maybe your reasoning isn't perfectly

mechanized yet and You should work harder at

Yourself.

Why don't you read up on Turing and the Unsolvability of

the Halting Problem for Turing machines ?

Because it's off topic.

Since the subject is: ``What leads us to say "We don't know" '' ,

I'd venture to say that the unsolvability of the Halting Problem

OK, let's try.

So, we have the halting problem.

And what leads us to say "we don't know"?

Godel's formalism states it clearly - there is

no implication between (a is proven) and a. Right?

Tell me, do You often say "we've proven that halting

problem is undecidable, so we can't know whether

it is or not"

?

Somehow I've never met a mathematician talking

this way.

Pretty good.

Suppose I speak only Mandarin Chinese and you and I are

on the phone.

Can you then make me understand what the Polish

word form napisał means?

No.

And, returning to the subject,

Godel's formalism states it clearly - there is

no implication between (a is proven) and a. Right?

Tell me, do You often say "we've proven that all

consistent theories [of some kind] are incomplete,

so we can't know whether they are or not"

?

Somehow I've never meet a mathematician talking

this way.

[...]

Goedel and others who think about incompleteness

study arbitrarily powerful First Order Logic

theories, with , in principle, an arbitrarily large

collection/set of axioms, and axiom schemas.

Logicians have studied formal theories or

large cardinal axioms that lead to

inconsistent theories (Frege and the

set of all sets...

From Wikipedia on Russell's Paradox:

" In a 1902 letter,[6] he announced the discovery to Gottlob Frege of

the paradox in Frege's 1879 Begriffsschrift and framed the problem in

terms of both logic and set theory, and in particular [...] "

please see:

< https://en.wikipedia.org/wiki/Russell%27s_paradox#History > .

Let's say mathematicians and logicians *hope* or

*believe* that ZFC set theory is consistent,

but have no expectation of proving consistency

in a simple manner.

If you polled mathematicians on whether they

KNOW/believe that ZFC set theory *is* consistent,

I think some would be cautious and say:

" I believe ZFC is consistent, but I can't say

that I can prove it ... ".

Does that seem on-topic?

David Bernier