Edward Frenkel

speaker

2,304 appearances 1 recordings 1 series first heard Apr 2023 last heard Apr 2023

Edward Frenkel’s voice in public audio — every appearance, attributed to the second.

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So every possibility can be realized.

There are different flavors.

This is a good illustration of what a formal system is.

You start with a set of axioms, those statements that you take for granted, and this is where you have a choice.

And by making different choices, you actually create different mathematics.

After that, there are rules of inference, logical rules, such as if A is true and A implies B, then B is true.

Most of them were actually introduced already by Aristotle, even before Euclid.

And then it runs as follows.

You have the axioms, which are accepted as true statements.

Then you have a way to produce new statements by using the rules of logical inference from the axioms.

Every statement you obtain, you call a theorem, and you kind of add it to the collection of true statements.

And then the question is, how far can you go?

How many statements can you prove this way?

Of course, you want the system to be non-trivial in the sense that you don't prove everything.

Because if you prove everything, it would mean that it's self-contradictory, that you prove a statement A and it's negation.

So that's kind of useless.

It has to be discriminating enough

so that it doesn't prove contradictory statements.

So there is already a question of that mathematical consistency.

It has to be consistent in the sense that it is not self-contradictory.

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