[FOM] PA inconsistencies
Sam Sanders
sasander at me.com
Sun Aug 19 15:39:56 EDT 2018
Dear José,
I am generally going to second what Tim wrote in his previous email, and emphasise his question:
> What does "factually true" mean? Is it "factually true" that the square root of 2 is irrational? What about Fermat's Last Theorem? If so, why are these factually true while other proven theorems are not?
There are a number of questions I have though:
> the word VERY is not operative. It do not think that 11 VERYs is an unbounded number.
So this use of VERY does indeed remind one of the (usual) sorites paradox as follows.
Consider the predicate P(x) defined as:
x+1 many VERYs is a stronger statement than x many VERYs.
Then P(1) and P(2) are true, but P(11) seems false, as you say.
A couple of questions:
1) How does the translation t manage to formalise this kind of sorites-like behaviour?
2) Such sorites phenomena are extremely sensitive to context; how does t manage that?
> Now, I will focus on the infinitely large strictly decreasing sequence of ordinals.
>
> Sam Sanders wrote: What would the significance be of such a sequence, when it plays a role in some theory of say physics?
>
> As I already mentioned, such a sequence is very similar to the sorites paradox: https://plato.stanford.edu/entries/sorites-paradox/
Similar how? The aforementioned sequence is infinite (by your own admission), while the sorites is usually about finite things.
(There is work on generalising the sorites, but that uses absurdly strong axioms, relative to the current discussion).
> Even if such a sequence does not exist in reality, real numbers neither
Please show me your evidence/proof that real numbers do not exist in reality.
> The word VERY is another example of a non-ZFC concept with applications in reality.
Please show me your evidence/proof that (something like) VERY cannot be developed in ZFC. Does it require large cardinals perhaps?
It is OK to prefer other foundations over ZFC (some of my best friends are like that), but grand claims need grand proofs/evidence.
Best,
Sam
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