I think you underestimate the zel of the 1000 top mathematicians about this infinity business. Perhaps there is about 10 dissidents among them, who are finitists.
But for all the others, the concept of infinity is a fundamental one. It’s easier to find a Catholic bishop who doesn’t care about God or Trinity, than a top mathematician who doesn’t believe and love infinity.
The infinity mythology is quite a rich one. Surprisingly rich, indeed.
First, we have the countable infinity—like all the natural numbers, or all the rational numbers. Aleph 0 is the cardinal of those. Then, we have 2^aleph0 or aleph1. The number of all real numbers. Between those two, you are free to postulate some greater then aleph0 and smaller than aleph1. You are also free to postulate that there is no cardinal between aleph zero and aleph one. It’s the Cohen’s solution of the so called Continuum problem.
You have more alephs above aleph 1. One aleph for every natural number. And you are free to postulate (or not) cardinal numbers between each two.
Then you have trans aleph big cardinal numbers.
Then you have non-constructible big infinite cardinals.
And then you have a lot of ordinals.
And then you have the Axiom of Choice. You can accept it or not. But it is proven, that if the whole structure of infinity is broken. then is broken with or without the Axiom of Choice.
I suspect Terence Tao has no strong opinion about ZF or ZFC system. He loves prime numbers and all the natural and real numbers and so on. But I don’t think he is in Set Theory very much.
I think you underestimate the zel of the 1000 top mathematicians about this infinity business. Perhaps there is about 10 dissidents among them, who are finitists.
But for all the others, the concept of infinity is a fundamental one. It’s easier to find a Catholic bishop who doesn’t care about God or Trinity, than a top mathematician who doesn’t believe and love infinity.
The infinity mythology is quite a rich one. Surprisingly rich, indeed.
First, we have the countable infinity—like all the natural numbers, or all the rational numbers. Aleph 0 is the cardinal of those. Then, we have 2^aleph0 or aleph1. The number of all real numbers. Between those two, you are free to postulate some greater then aleph0 and smaller than aleph1. You are also free to postulate that there is no cardinal between aleph zero and aleph one. It’s the Cohen’s solution of the so called Continuum problem.
You have more alephs above aleph 1. One aleph for every natural number. And you are free to postulate (or not) cardinal numbers between each two.
Then you have trans aleph big cardinal numbers.
Then you have non-constructible big infinite cardinals.
And then you have a lot of ordinals.
And then you have the Axiom of Choice. You can accept it or not. But it is proven, that if the whole structure of infinity is broken. then is broken with or without the Axiom of Choice.
I suspect Terence Tao has no strong opinion about ZF or ZFC system. He loves prime numbers and all the natural and real numbers and so on. But I don’t think he is in Set Theory very much.