Mathematicians who are finitists.

[Jehanne]

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.

[polymath257]

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.

Hmmm....there can be no actual proof of the existence of an infinite set because the collection of hereditarily finite sets is a perfectly good model of set theory without the axiom of infinity. That shows the internal consistency of the axioms. Again, in the finitist system, the *collection* of rational numbers isn't something that can be constructed. All that can be done is saying whether particular constructs are rational numbers or not.

The point is that finitistic math has very strict rules for set formation and those rules do NOT allow talking about any infinite sets.

Now, this is not to say that allowing infinite sets isn't better in many ways. It gives wonderful insights into a large variety of phenomena. It makes many arguments much easier and produces results that finitistic reasoning cannot, even results concerning finite sets.

[Jehanne]

Hmmm....there can be no actual proof of the existence of an infinite set because the collection of hereditarily finite sets is a perfectly good model of set theory without the axiom of infinity. That shows the internal consistency of the axioms. Again, in the finitist system, the *collection* of rational numbers isn't something that can be constructed. All that can be done is saying whether particular constructs are rational numbers or not.

The point is that finitistic math has very strict rules for set formation and those rules do NOT allow talking about any infinite sets.

Now, this is not to say that allowing infinite sets isn't better in many ways. It gives wonderful insights into a large variety of phenomena. It makes many arguments much easier and produces results that finitistic reasoning cannot, even results concerning finite sets.

Here's a proof by John Hopcroft, Jeffrey Ullman and Rajeev Motwani in their Introduction to Automata Theory, Languages, and Computation Edition 3 textbook, page 9:

[polymath257]

Hmmm....there can be no actual proof of the existence of an infinite set because the collection of hereditarily finite sets is a perfectly good model of set theory without the axiom of infinity. That shows the internal consistency of the axioms. Again, in the finitist system, the *collection* of rational numbers isn't something that can be constructed. All that can be done is saying whether particular constructs are rational numbers or not.

The point is that finitistic math has very strict rules for set formation and those rules do NOT allow talking about any infinite sets.

Now, this is not to say that allowing infinite sets isn't better in many ways. It gives wonderful insights into a large variety of phenomena. It makes many arguments much easier and produces results that finitistic reasoning cannot, even results concerning finite sets.

Here's a proof by John Hopcroft, Jeffrey Ullman and Rajeev Motwani in their Introduction to Automata Theory, Languages, and Computation Edition 3 textbook, page 9:

Sorry, but it doesn't prove what you claimed. This proposition says that if S is finite and U is infinite, then T=U-S,
the complement of S in U, is infinite.

This does NOT prove the existence of an infinite set. It shows that *if* there is an inifnite set U, then there are other
infinite sets (U-S where S is finite).

Now, show how to get that set U from finitist principles.

[Jehanne]

Here's a proof by John Hopcroft, Jeffrey Ullman and Rajeev Motwani in their Introduction to Automata Theory, Languages, and Computation Edition 3 textbook, page 9:

Sorry, but it doesn't prove what you claimed. This proposition says that if S is finite and U is infinite, then T=U-S,
the complement of S in U, is infinite.

This does NOT prove the existence of an infinite set. It shows that *if* there is an inifnite set U, then there are other
infinite sets (U-S where S is finite).

Now, show how to get that set U from finitist principles.

Proofs that there are infinitely many primes

[polymath257]

Sorry, but it doesn't prove what you claimed. This proposition says that if S is finite and U is infinite, then T=U-S,
the complement of S in U, is infinite.

This does NOT prove the existence of an infinite set. It shows that *if* there is an inifnite set U, then there are other
infinite sets (U-S where S is finite).

Now, show how to get that set U from finitist principles.

Proofs that there are infinitely many primes

And the finitist would not accept that there is a set of primes. Instead, they would say that for every finite set of primes, there is a prime not in that set.

For example, the orignal proof by Euclid was of this form. And  by modern standards, Euclid would have been a finitist. The topological proof would certainly NOT be accepted by finitists. Some formulation of the Goldbach proof would probably be acceptable (again, for any finite set of primes, there is a prime not in the list). Kummer *was* a finitist.

Once again, for a finitist, NO infinite collection is acceptable: only finite collections.

It is possible to talk about the *property* of being a natural number, but not the *set* of all such numbers. It is possible to talk about the *property* of primality, but not the set of all such. So, a finitist would simply say there is no 'collection of all primes'.

But this is similar to a modern set theorist saying there is no 'collection of all sets' since it is provably the case that this collection is not a set.

[Jehanne]

Once again, for a finitist, NO infinite collection is acceptable: only finite collections.

Polymath,

You have been such a blessing!

One last question, "Do finitists make any distinction between sets ("collections") that are countable versus uncountable?"

Thanks,

Dawn

[polymath257]

Once again, for a finitist, NO infinite collection is acceptable: only finite collections.

Polymath,

You have been such a blessing!

One last question, "Do finitists make any distinction between sets ("collections") that are countable versus uncountable?"

Thanks,

Dawn

Again, for a finitist, all sets are finite. So the issue simply doesn't arise. The definition of cardinality can be made, but it trivializes. No set is uncountable, no set is even countably infinite.

For a finitist, there *is* no set of natural numbers. There *is* no set of real numbers. ONLY finite sets exist in the finitist system.

I think you can see why most mathematicians see finitism as limiting.

[Jehanne]

Polymath,

You have been such a blessing!

One last question, "Do finitists make any distinction between sets ("collections") that are countable versus uncountable?"

Thanks,

Dawn

Again, for a finitist, all sets are finite. So the issue simply doesn't arise. The definition of cardinality can be made, but it trivializes. No set is uncountable, no set is even countably infinite.

For a finitist, there *is* no set of natural numbers. There *is* no set of real numbers. ONLY finite sets exist in the finitist system.

I think you can see why most mathematicians see finitism as limiting.

I found Cantor's diagonalization proofs, as described by Rosen, to be on par with the Pythagorean theorem. Seems like the finistists are engaging in ad hoc reasoning. But, maybe they feel that there are really square circles?

[polymath257]

Again, for a finitist, all sets are finite. So the issue simply doesn't arise. The definition of cardinality can be made, but it trivializes. No set is uncountable, no set is even countably infinite.

For a finitist, there *is* no set of natural numbers. There *is* no set of real numbers. ONLY finite sets exist in the finitist system.

I think you can see why most mathematicians see finitism as limiting.

I found Cantor's diagonalization proofs, as described by Rosen, to be on par with the Pythagorean theorem.  Seems like the finistists are engaging in ad hoc reasoning.  But, maybe they feel that there are really square circles?

I have one book (The Foundations of Mathematics by Kunen) that discusses the Philosophical differences between Platonists, Finitists, and Formalists as being similar to those who are theists, atheists, and agnostics. The Platonists believe in the existence of mathematical objects in some real sense. Finitists limit themselves to only finite sets. And formalists organize their lives so that the question isn't particularly relevant.

So, for the Platonist, the Continuum Hypothesis is either true or false. Our axiom system may not be strong enough to answer the question of which, but for the Platonist there is a definite answer.

For the finitist, the Continuum Hypothesis is simply meaningless. It talks about the sizes of different infinite sets and, for the finitist, this is literally meaningless.

The formalist is happy with any axiom system that gives nice results. The Continuum Hypothesis has no answer beside whatever axioms we *choose* that might resolve it.

For the Platonist, the set of real numbers is really a larger cardinality than that of the natural numbers. For the formalist, this statement can be proved in some systems and not in others. And for the finitist, it is just meaningless.