What can we gain by losing infinity?

Which is exactly why I mentioned rejection of zero, negative numbers, etc.

You can reject them, but doing so just throws away useful tools without gaining anything in return.

When I was about ten, a math teacher once asked me whether the number 0.9999... (infinitely repeating) was different than 1. I said, with my child's intuition, that of course it was. He then challenged me to write down a number that was in between them, because if they were not the same number then there would be many (in fact, infinitely many) numbers between them. I couldn't, of course: the best I could do was to write 0.9999...5, which falls into the same category error as "infinity plus one / infinity plus two".

Now, decades later, I get it better. The number 0.99999... is 9/10 + 9/100 + 9/1000 + 9/10000 + ..., which approaches 1 asymptotically the same way that 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... approaches 1. Under many circumstances, you can treat that number as if it was 1, which neatly answers Zeno's Paradox. (Though beware of the limitations of that analysis: 1/n approaches infinity as n approaches 0, but 1/0 is not equal to infinity. Because 1/n approaches infinity only as n approaches 0 from the positive direction. If you look at the sequence 1/-0.1, 1/-0.01, 1/-0.001, etc. where n approaches 0 from the negative direction, that approaches negative infinity. A function that has two different limits as you approach the same number from two different directions cannot have its limit substituted like that).

> But in the late 1800s, Georg Cantor and other mathematicians showed that the infinite really can exist.

I think, as I understand it, the objection is this. The proposition that infinity is "real", and there are actually infinite (not just very many) things.

Contrarian thinking can be great because it taps into the intuition that the masses are mostly followers who can be led anywhere, not critical thinkers who've deeply examined what they believe. Being contrarian, then, is akin to staking out a new leadership position.

The space of contrarian ideas is vast, and most of them are probably bad, but, nevertheless, the willingness to hold unconventional, internally consistent views should be celebrated, because it increases diversity of thought. Our collective hive mind grows stronger through heresy.

However, I like my heresy with a splash of axiomatic precision, which is sadly lacking in this article.

> The article doesn’t really tell us what is gained by rejecting infinity.

Decidability. The issues around undecidability all involve the lack of an upper bound. In a finite deterministic space, everything is decidable, although some things may be too costly computationally to decide.

There are several ways to go for decidability. The brute force way is computer arithmetic - there is no number larger than 2^64-1. That's how we get things done on computers, but proofs about numbers with finite upper bounds need lots of special cases. Mathematicians hate that.

I used to work on this sort of thing, using Boyer-Moore theory. That's a lot like the Peano axioms. There is (ZERO), and (ADD1 (ZERO)), and (ADD1 (ADD1 (ZERO))), etc. Everything is constructive and has an unambiguous representation in a LISP-like form. You can have recursive functions. But they must be proven to terminate, by having a nonnegative value which decreases on each recursive call. There is a distinction between "infinite" and "arbitrarily large". You can talk about arbitrarily large numbers, but you cannot get to 1/2 + 1/4 + 1/8 ... = 1. You can have integers and rational numbers of arbitrary size, but not reals.

Set theory was interesting. Rather than axiomatic set theory, I was using lists as sets, with the constraints that no value could be duplicated and the list must be ordered. Equality is strict - two things are equal only if the elements are all equal, compared element by element. It's possible to prove the usual axioms of set theory via this route. The ordered criterion requires proving things about ordered list insertion to get there. It's ugly and needs machine proofs.

I was doing this back in the early 1980s, when machine proofs were frowned upon. Mathematicians were still upset about the four-color theorem proof. It's all case analysis, with thousands of cases. That's more acceptable today.

Looked at in this light, infinity is a labor-saving device to eliminate special cases, at a potential cost in soundness.