Re: Torkel Franzen on truth

Newberry <newberryxy@xxxxxxxxx> writes:

On Dec 8, 2:04 pm, Alan Smaill <sma...@xxxxxxxxxxxxxxxx> wrote:
Newberry <newberr...@xxxxxxxxx> writes:
The cornerstone of TF's argument is that the consistency of ZFC is
provable in ZFC + an axiom of infinity, which is no longer manifestly
true. We hesitate as we go higher up in the chain of theories, hence -
according to TF - we are not better than any machine because we cannot
say about an arbitrary system that its Goedel formula is true. Yet he
is absolutely sure that PA and ZFC are consistent.

Where did TF claim that he was "absolutely sure that ... ZFC [is]
consistent"?

I see no evidence that that was his view in the cite you posted
earlier in the thread.

He has an entire chapter in his book "Skepticism and Confidence",
where he refutes the skeptics.

He has more than one book; do you mean in
"Gödel's Theorem: An Incomplete Guide to its Use and Abuse"?

"And given that the axioms of ZFC are so utterly compelling, so
obviously true in the world of sets, we can do no better than to adopt
these axioms as our starting point. Since the axioms are true, they
are also consistent." [p.105]

I'll look at the context before commenting, since that is very different
from his attitude to ZFC in "Inexhausibility".

"if the axioms of ZFC are manifestly true, they are obviously
consistent" [p.105]

Clearly that is a hypothetical statement, which does not commit him to
the antecedent.

But it does not matter if TF is convinced that ZFC is consistent. I
claim that there are only three possibilities:
1) We do not know if PA's Goedel sentence is true. So we do not know
that the set of truth is productive.

That's a bunch of stuff you have already bundled up --
the 'so' part is not a logical consequence of the the first sentence,
for a start.

2) The human mind surpasses any machine
3) There axists a formalization of arithmetic that can prove its own
consistency

Thus far I have not seen any convincing argument that we can reject
all three. In fact based on what I have seen I am convinced that we
cannot.

Are you opting for 1)?

Maybe you can clarify what you are asking about:
that we do not know that PA's Goedel sentence is true;
that we do not know if the set of truths is productive;
or that the second follows from the first?

You are a second convert. Daryl is now also
leaning in that direction. It would make more sense than attempting to
argue that we can reject all three, but it is quite a departure from
what most logicians believe. Peter Smith will certainly differ.



--
Alan Smaill
.

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