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The first recorded use of a specific word for the infinite is credited to Anaximander around 580BCE. His term apeiron can be translated into several ways - boundless, limitless, indefinite, infinite. His context was a search for the origins of all things, which he held to be an endless primordial mass. Being inexhaustable, aperion could generate everything in existence without being used up. Many scholars believed it to be a primeval chaos that can be separated into the four Greek elements - earth, air, fire, water, from which all else was formed.
… our minds seem to require the idea that things might go on forever - in space and time and the future and the past. Infinity is perhaps a natural default to our pattern seeking minds.
Infinity is bigger than any specific whole number, whatever notational system we use and whatever new names we invent. In practice we run out of names before we run out numbers.
Archimedes grains of sand: he concluded that in our terms, at most 10^63 grains of sand can fill the universe. With today’s figures for the size of the observable universe that number becomes 10^93 (still finite).
Aquinas - an infinite change of causality is impossible, so there must be a first cause. This is god.
Today’s mathematicians don’t consider the distinction between actual and potential infinity to be important, because mathematical objects are only actual on a conceptual level. What about ‘two’? I can’t show you the number ‘two’ but but I can hold up two cats, or two chairs. Holding up a piece of paper with a ‘2’ on it is a symbol for the number, not the actual number.
All numbers ever used are smaller that some specified number. So in practice only numbers smaller than that bound have ever been needed. Absolutely no activities that depend on numbers, in the whole of human history would change if we limited ourselves to this finite range of numbers.
NOTE
A human can be easily tricked. I have not made any attempts at climbing towards outerspace. I just accept it is impossible. My own lived in system, for all its adventures and hopes, is small. I can just be in an earth sized room, or a town size room and not know. Traveling home, I’ll pass many exits, none of which I’ll take. They lead to small towns, of people probing their finiteness or this is what we see.
I should each day take a new way home, till exhausting my travels I’ll wrap around the island.
Some people feel caged in their existence so they go out. I understand this personally, but it also does feel like a surface level of exploration. You get on a plane, land someplace different, and perform similar actions you would elsewhere in the world. Imagine doing that and being bored.
It’s why the style of social media tourism feels so hollow. The actual depth comes from intellectually engaging with the new environment, or sharing the experience. This feels a further infinity.
Wittgenstein despised the diagonal argument. With Cantor long dead Witt. continued to express his profound philosophical dissatisfaction through the through the pernicious idioms of ‘set theory’