Would the proof of consistency from the other set of axioms rely on the consistency of those axioms, which in turn relies on a proof of consistency from a further set of axioms, and so on? How do you get around this? Is it even an issue? Could you prove just that based on your axioms for the...
Another question, how can you be sure that a system of axioms does not lead to a contradiction when you develop the theory from them? E. G. Is it possible that the Peano axioms could lead to a contradiction? As you've probably guessed I'm a bit confused in general...
Sorry if that title doesn't match up well with my question. I think it captures roughly what I'm wondering about.
My uncertainty is to do with how much of mathematics is certainly true. Like, if I picked up any undergrad or grad textbook in mathematics, would everything in it be true based on...
Are there any books or a sequence of books I could study to build up a picture of what I can know for sure and why I can know it? I suppose that's what some people do when they study a subject and move from the basic principles to further stuff.
So like, has anybody started with the smallest...