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.
In the language of model theory, Nelson believes that one can prove that Peano arithmetic has no models. He does not believe one can prove that Peano arithmetic minus induction has no models.
That… is a very strange combination of beliefs. I honestly cannot imagine what he could possibly be thinking. It’s like someone sat around thinking, “Hm, what could I come up with that would be even dumber than believing PA to be inconsistent?” Indeed, I notice my own confusion and suspect that you may have misunderstood something that was stupid but not quite that stupid.
But we do not have to have a model, or even to know any model theory, to “talk about something.”
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.
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.
That… is a very strange combination of beliefs. I honestly cannot imagine what he could possibly be thinking. It’s like someone sat around thinking, “Hm, what could I come up with that would be even dumber than believing PA to be inconsistent?” Indeed, I notice my own confusion and suspect that you may have misunderstood something that was stupid but not quite that stupid.
Then what is it talking about?
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.