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Mathematicians who are finitists.
#31
RE: Mathematicians who are finitists.
(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.
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Messages In This Thread
Mathematicians who are finitists. - by Jehanne - April 2, 2019 at 8:49 am
RE: Mathematicians who are finitists. - by Fireball - April 2, 2019 at 10:08 am
RE: Mathematicians who are finitists. - by Jehanne - April 3, 2019 at 12:10 pm
RE: Mathematicians who are finitists. - by Smaug - April 4, 2019 at 4:49 pm
RE: Mathematicians who are finitists. - by Jehanne - April 5, 2019 at 3:14 pm
RE: Mathematicians who are finitists. - by Smaug - April 5, 2019 at 4:04 pm
RE: Mathematicians who are finitists. - by chimp3 - April 4, 2019 at 5:43 pm
RE: Mathematicians who are finitists. - by Sal - April 5, 2019 at 3:55 pm
RE: Mathematicians who are finitists. - by Jehanne - April 6, 2019 at 9:26 am
RE: Mathematicians who are finitists. - by Smaug - April 6, 2019 at 4:33 pm
RE: Mathematicians who are finitists. - by Jehanne - April 6, 2019 at 10:46 pm
RE: Mathematicians who are finitists. - by Smaug - April 7, 2019 at 1:49 pm
RE: Mathematicians who are finitists. - by Jehanne - April 7, 2019 at 3:52 pm
RE: Mathematicians who are finitists. - by Smaug - April 7, 2019 at 5:25 pm
RE: Mathematicians who are finitists. - by Jehanne - April 8, 2019 at 6:08 pm
RE: Mathematicians who are finitists. - by Little lunch - April 7, 2019 at 5:50 pm
RE: Mathematicians who are finitists. - by polymath257 - April 8, 2019 at 9:56 pm
RE: Mathematicians who are finitists. - by Fireball - April 9, 2019 at 10:32 am
RE: Mathematicians who are finitists. - by robvalue - April 9, 2019 at 6:18 am
RE: Mathematicians who are finitists. - by polymath257 - April 9, 2019 at 6:54 pm
RE: Mathematicians who are finitists. - by Jehanne - April 9, 2019 at 9:46 pm
RE: Mathematicians who are finitists. - by polymath257 - April 10, 2019 at 8:42 am
RE: Mathematicians who are finitists. - by Jehanne - April 10, 2019 at 8:59 am
RE: Mathematicians who are finitists. - by polymath257 - April 11, 2019 at 10:48 am
RE: Mathematicians who are finitists. - by Jehanne - April 11, 2019 at 11:09 am
RE: Mathematicians who are finitists. - by polymath257 - April 11, 2019 at 11:24 am
RE: Mathematicians who are finitists. - by Jehanne - April 11, 2019 at 2:13 pm
RE: Mathematicians who are finitists. - by polymath257 - April 11, 2019 at 3:33 pm
RE: Mathematicians who are finitists. - by Jehanne - April 11, 2019 at 3:39 pm
RE: Mathematicians who are finitists. - by polymath257 - April 11, 2019 at 5:17 pm
RE: Mathematicians who are finitists. - by Jehanne - April 14, 2019 at 7:15 am
RE: Mathematicians who are finitists. - by polymath257 - April 14, 2019 at 9:31 am
RE: Mathematicians who are finitists. - by Jehanne - April 14, 2019 at 12:07 pm
RE: Mathematicians who are finitists. - by polymath257 - April 14, 2019 at 1:38 pm
RE: Mathematicians who are finitists. - by Fireball - April 9, 2019 at 10:03 pm
RE: Mathematicians who are finitists. - by Jehanne - April 9, 2019 at 11:05 am
RE: Mathematicians who are finitists. - by robvalue - April 9, 2019 at 1:22 pm
RE: Mathematicians who are finitists. - by Sal - April 9, 2019 at 9:28 pm
RE: Mathematicians who are finitists. - by polymath257 - April 9, 2019 at 9:44 pm
RE: Mathematicians who are finitists. - by polymath257 - April 15, 2019 at 9:58 am
RE: Mathematicians who are finitists. - by LastPoet - April 15, 2019 at 12:32 pm
RE: Mathematicians who are finitists. - by polymath257 - April 15, 2019 at 5:06 pm
RE: Mathematicians who are finitists. - by LastPoet - April 16, 2019 at 12:58 pm
RE: Mathematicians who are finitists. - by Fireball - April 16, 2019 at 3:25 pm
RE: Mathematicians who are finitists. - by polymath257 - April 16, 2019 at 5:43 pm
RE: Mathematicians who are finitists. - by polymath257 - June 29, 2019 at 6:44 pm
RE: Mathematicians who are finitists. - by Jehanne - June 30, 2019 at 12:03 pm
RE: Mathematicians who are finitists. - by LastPoet - June 30, 2019 at 1:30 pm
RE: Mathematicians who are finitists. - by Jehanne - June 30, 2019 at 3:34 pm
RE: Mathematicians who are finitists. - by polymath257 - July 1, 2019 at 2:02 pm
RE: Mathematicians who are finitists. - by Jehanne - July 2, 2019 at 7:50 am
RE: Mathematicians who are finitists. - by Jehanne - July 2, 2019 at 11:36 am
RE: Mathematicians who are finitists. - by Jehanne - July 2, 2019 at 11:29 pm
RE: Mathematicians who are finitists. - by Jehanne - July 3, 2019 at 11:51 am
RE: Mathematicians who are finitists. - by Jehanne - July 3, 2019 at 10:06 pm
RE: Mathematicians who are finitists. - by Jehanne - July 4, 2019 at 7:33 am
RE: Mathematicians who are finitists. - by polymath257 - July 4, 2019 at 11:24 am
RE: Mathematicians who are finitists. - by Fireball - June 29, 2019 at 10:00 pm
RE: Mathematicians who are finitists. - by LastPoet - July 1, 2019 at 2:38 pm
RE: Mathematicians who are finitists. - by polymath257 - July 2, 2019 at 1:45 pm
RE: Mathematicians who are finitists. - by Jehanne - July 4, 2019 at 6:46 pm
RE: Mathematicians who are finitists. - by Jehanne - July 5, 2019 at 6:36 am
RE: Mathematicians who are finitists. - by Jehanne - July 5, 2019 at 9:09 am
RE: Mathematicians who are finitists. - by polymath257 - July 7, 2019 at 3:38 am
RE: Mathematicians who are finitists. - by Jehanne - July 8, 2019 at 7:05 am
RE: Mathematicians who are finitists. - by Jehanne - July 8, 2019 at 3:38 pm
RE: Mathematicians who are finitists. - by Jehanne - July 9, 2019 at 7:51 am
RE: Mathematicians who are finitists. - by polymath257 - July 9, 2019 at 9:56 am
RE: Mathematicians who are finitists. - by polymath257 - July 10, 2019 at 4:55 am
RE: Mathematicians who are finitists. - by Jehanne - July 10, 2019 at 7:55 am
RE: Mathematicians who are finitists. - by polymath257 - July 10, 2019 at 8:34 am
RE: Mathematicians who are finitists. - by Jehanne - July 10, 2019 at 8:19 pm
RE: Mathematicians who are finitists. - by polymath257 - July 11, 2019 at 2:52 am
RE: Mathematicians who are finitists. - by Jehanne - July 11, 2019 at 9:30 am
RE: Mathematicians who are finitists. - by polymath257 - July 11, 2019 at 10:20 am
RE: Mathematicians who are finitists. - by LastPoet - July 5, 2019 at 12:16 pm
RE: Mathematicians who are finitists. - by Jehanne - July 5, 2019 at 1:38 pm
RE: Mathematicians who are finitists. - by Jehanne - July 9, 2019 at 2:29 pm

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