↑ comment by Viliam ·
2016-12-21T14:12:54.770Z · LW(p) · GW(p)
Okay, I am happy we came to an agreement, but...
But tell this to the 99% of the top mathematicians! For them all those infinities - infinite multitude of them - exist.
...I wonder whether this is true. I mean, as long as someone knows that you can have multiple mutually incompatible systems of axioms, then they should on some level realize that the choice of a specific system is arbitrary.
I mean, I haven't realized this for a long time simply because I read have some popular books about infinities probably already at elementary school age, and they didn't start with a big red warning "everything you will read in this book is merely a consequence of an arbitrarily selected set of axioms; other sets of axioms could lead to completely different outcomes; and the choice of the axioms is arbitrary because nothing of this is real, mwa-ha-ha-ha!" So, I naively assumed it was just business as usual. If I didn't suspect books about prime numbers or complex numbers, why should I suspect the books about infinities?
And when I later heard something about the axioms, the idea of the "one true system of aleph-zero, aleph-one, etc." was already firmly stuck in my head. So even while I was reading about how different axioms are possible, I still on some level assumed that the ones I was already familiar with are -- in some sense -- the correct ones (without asking myself what exactly the word "correct" could even possibly mean in this context).
Only a few years later I have finally connected the dots in my head. It was while thinking about natural numbers, the fifth Peano's axiom, how without it we could have different models of natural numbers, and what exactly that means. And a few years later, my friend complained about the axiom of choice, and suddenly something clicked in my head, and I was like "wait, isn't that the same thing as with the fifth Peano's axiom? like you can have different models that all satisfy the same set of axioms, but they still differ at some underspecified places, and you need an extra axiom to distinguish between them?" Because at that moment I already had an understanding of what "having different models" means; it just never occured to me to apply it to infinities. (Despite seeing some hints which in hindsight obviously tried telling me to do so.)
In my defense, though, I don't remember studying about infinities at university. Probably not even at high school, or at least not so deeply. So all my knowledge of infinities came from the popular books I have read at childhood; it shouldn't be surprising it was seriously incomplete. I assume that people who actually studied this stuff at university, who were explicitly told about the existence of various axiomatic systems and given specific examples thereof, and then had exercises and exams... I believe they were in much better position to make the connection. So it would surprise me a lot if the best of them just couldn't connect the dots even better than I did.
On the other hand, sometimes even highly intelligent and educated people fail to connect the dots.
I admit I am somewhat tempted to write an e-mail to Terence Tao and ask him what he thinks about this. There was never a better excuse to bother people much smarter than me. :D
Replies from: Thomas
↑ comment by Thomas ·
2016-12-21T15:46:30.769Z · LW(p) · GW(p)
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.