If you can prove that a theory has no models, then you can prove a contradiction in a finite number of steps.
No, if a first-order theory has no models, then you can prove a contradiction from it. Not, if it provably has no models. Just, if it has no models, period, then it proves a contradiction in a finite number of steps.
What I said is a true consequence of what you said, so why are you frustrated? I am trying to make statements that a formalist (such as Nelson) would endorse.
That… is a very strange combination of beliefs. I honestly cannot imagine what he could possibly be thinking.
Now you’re confusing me. Induction does not follow from the other axioms, so one does not have to reject all of Peano arithmetic to reject induction. Why is it more stupid to reject only induction?
You might reply: because P(everything is wrong | induction is wrong) is large. (Though then you would be falling for the conjunction fallacy, which is something you would never do.) A lot of Nelson’s work can be seen as arguing that it is not as large as you think.
Then what is it talking about?
It is very rare for someone speaking about numbers to be talking about a particular model of numbers inside a particular model of set theory. The very word “model” is chosen to contrast it with “the real thing.”
Of course formalists reject the idea that there is a real thing.
What I said is a true consequence of what you said, so why are you frustrated? I am trying to make statements that a formalist (such as Nelson) would endorse.
Now you’re confusing me. Induction does not follow from the other axioms, so one does not have to reject all of Peano arithmetic to reject induction. Why is it more stupid to reject only induction?
You might reply: because P(everything is wrong | induction is wrong) is large. (Though then you would be falling for the conjunction fallacy, which is something you would never do.) A lot of Nelson’s work can be seen as arguing that it is not as large as you think.
It is very rare for someone speaking about numbers to be talking about a particular model of numbers inside a particular model of set theory. The very word “model” is chosen to contrast it with “the real thing.”
Of course formalists reject the idea that there is a real thing.