(April 11, 2019 at 10:48 am)polymath257 Wrote:(April 10, 2019 at 8:59 am)Jehanne Wrote: They have tapes of infinite length, though, at least abstractly:

Wikipedia -- Turning machine

Do finitists reject Cantor's diagonalization proofs, namely, that Aleph-naught is the smallest infinite set, followed by Aleph-one, etc.?

(Sorry to be asking you this, but I don't know of any finitists whom I can ask!)

Well, strict finitists only have finite sets, so the question of the sizes of infinite sets simply doesn't arise. So, individual numbers can be considered, any any finite set of numbers, but the collection of all natural numbers is rejected. Since rational numbers are essentially reduced pairs of natural numbers (think fractions), a finitist can also talk about natural numbers. But it becomes much harder to even talk about real numbers, let alone the collection of *all* real numbers. So the diagonalization argument, as usually seen, doesn't abide by finitist principles.

They will admit the possibility of adding a new element to any already existing set, but not to allow the union over all such processes to get an actual infinite set.

It is possible to formulate Turing machines in such a way that the tape is only *potentially* infinite as opposed to actually infinite. The idea is that a new cell is added at either end if required. In this way, the collection of cells is finite at every step in time. This is how a typical finitist would speak of Turing machines. And, again, this is their bread and butter--finite state machines, recursive functions, etc.

The fact that the set of rational numbers are countable and the set of real numbers is uncountable is proof that actual infinities, at least in the abstract sense, are completely coherent mathematically. Cantor demonstrated this beyond any shadow of doubting reality. I am no expert, but have read Rosen multiple times over the years, now in its 8th edition, having first taken Discrete Mathematics with his 1st edition back in 1990. Aho, Uullman and Hopcroft also have a proof of the existence of infinite sets that I will try to post here.

And without delay Peter went quickly out of the synagogue (assembly) and went unto the house of Marcellus, where Simon lodged: and much people followed him...And Peter turned unto the people that followed him and said: Ye shall now see a great and marvellous wonder. And Peter seeing a great dog bound with a strong chain, went to him and loosed him, and when he was loosed the dog received a man's voice and said unto Peter: What dost thou bid me to do, thou servant of the unspeakable and living God? Peter said unto him: Go in and say unto Simon in the midst of his company: Peter saith unto thee, Come forth abroad, for thy sake am I come to Rome, thou wicked one and deceiver of simple souls. And immediately the dog ran and entered in, and rushed into the midst of them that were with Simon, and lifted up his forefeet and in a loud voice said: Thou Simon, Peter the servant of Christ who standeth at the door saith unto thee: Come forth abroad, for thy sake am I come to Rome, thou most wicked one and deceiver of simple souls. And when Simon heard it, and beheld the incredible sight, he lost the words wherewith he was deceiving them that stood by, and all of them were amazed. (The Acts of Peter, 9)