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Mathematicians who are finitists.
#11
RE: Mathematicians who are finitists.
(April 6, 2019 at 4:33 pm)Smaug Wrote: However, it's still unclear for me what is the principal difference between potential and actual infinity other than latter being, to put it simple, an 'end cap' of a set.

An actual infinity means a set (say, the natural numbers) has a cardinality that is infinite, as opposed to a "potential infinite", which is always finite, even given the fact such "grows" forever.
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#12
RE: Mathematicians who are finitists.
(April 6, 2019 at 10:46 pm)Jehanne Wrote:
(April 6, 2019 at 4:33 pm)Smaug Wrote: However, it's still unclear for me what is the principal difference between potential and actual infinity other than latter being, to put it simple, an 'end cap' of a set.

An actual infinity means a set (say, the natural numbers) has a cardinality that is infinite, as opposed to a "potential infinite", which is always finite, even given the fact such "grows" forever.

Isn't it so that if we invoke the term 'cardinality' we imply that infinite sets exist? Saying that a set can be continued forever is basically saying it's infinite, isn't it?
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#13
RE: Mathematicians who are finitists.
(April 7, 2019 at 1:49 pm)Smaug Wrote:
(April 6, 2019 at 10:46 pm)Jehanne Wrote: An actual infinity means a set (say, the natural numbers) has a cardinality that is infinite, as opposed to a "potential infinite", which is always finite, even given the fact such "grows" forever.

Isn't it so that if we invoke the term 'cardinality' we imply that infinite sets exist? Saying that a set can be continued forever is basically saying it's infinite, isn't it?

It's a subtle distinction.   Virtually all mathematicians would say that the set of prime numbers is an actual infinite, whose cardinality is identical to the set of natural numbers; most philosophers would agree with this, also.  However, the set of real numbers (both rational and irrational) is, as Cantor proved, an infinite set whose cardinality is greater than the set of natural numbers, even though both are infinite sets.  And, so, some infinities are bigger than others.  The set of natural numbers (and, prime numbers, as well as rational numbers) is a countably infinite set whereas the set of real numbers is uncountable.
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#14
RE: Mathematicians who are finitists.
(April 7, 2019 at 3:52 pm)Jehanne Wrote:
(April 7, 2019 at 1:49 pm)Smaug Wrote: Isn't it so that if we invoke the term 'cardinality' we imply that infinite sets exist? Saying that a set can be continued forever is basically saying it's infinite, isn't it?

It's a subtle distinction.   Virtually all mathematicians would say that the set of prime numbers is an actual infinite, whose cardinality is identical to the set of natural numbers; most philosophers would agree with this, also.  However, the set of real numbers (both rational and irrational) is, as Cantor proved, an infinite set whose cardinality is greater than the set of natural numbers, even though both are infinite sets.  And, so, some infinities are bigger than others.  The set of natural numbers (and, prime numbers, as well as rational numbers) is a countably infinite set whereas the set of real numbers is uncountable.

What I meant is that if we consider all sets to be fundamentally finite as finitists suggest then there's no need for such a term as cardinality. You can just explicitly specify the number of elements for any given set. And I still can't grasp the finitists' point of view, how they beat the "+1" arguement. For me such recursive definition automatically leads to the notion of infinity. Infinity makes perfect sense from practical point of view, too. In a non-rigorous approach it can be interpreted as a value which is out of scale.
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#15
RE: Mathematicians who are finitists.
I suppose that if something is growing forever, it would never be infinite until it stops growing.
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#16
RE: Mathematicians who are finitists.
(April 7, 2019 at 5:50 pm)Little lunch Wrote: I suppose that if something is growing forever, it would never be infinite until it stops growing.
Is it bad when smoke comes out of your ears?

If it stopped growing it wouldn't be infinite, infinite means it never will !!!
Infinity cannot be imagined or at least I have never succeeded, It's simply something i accept.
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#17
RE: Mathematicians who are finitists.
(April 7, 2019 at 5:25 pm)Smaug Wrote:
(April 7, 2019 at 3:52 pm)Jehanne Wrote: It's a subtle distinction.   Virtually all mathematicians would say that the set of prime numbers is an actual infinite, whose cardinality is identical to the set of natural numbers; most philosophers would agree with this, also.  However, the set of real numbers (both rational and irrational) is, as Cantor proved, an infinite set whose cardinality is greater than the set of natural numbers, even though both are infinite sets.  And, so, some infinities are bigger than others.  The set of natural numbers (and, prime numbers, as well as rational numbers) is a countably infinite set whereas the set of real numbers is uncountable.

What I meant is that if we consider all sets to be fundamentally finite as finitists suggest then there's no need for such a term as cardinality. You can just explicitly specify the number of elements for any given set. And I still can't grasp the finitists' point of view, how they beat the "+1" arguement. For me such recursive definition automatically leads to the notion of infinity. Infinity makes perfect sense from practical point of view, too. In a non-rigorous approach it can be interpreted as a value which is out of scale.

Potential infinite = always finite, even if it "grows" forever.

Actual infinite = a set of things (real or abstract) that already, right now, contains an infinite number of elements.
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#18
RE: Mathematicians who are finitists.
As a formalist in my mathematical philosophy, finitism is a possible way to do mathematics. But I find it incredibly limiting.

Here is the basic idea: only finite sets are allowed (actually, hereditarily finite sets). Nonetheless, it is possible to define what it means for a set to be a (Von Neuman) ordinal. For finite sets, this is equivalent to being a natural number. So finitists can talk about natural numbers and finite sets of natural numbers, but there is no way to talk about the set of *all* natural numbers.

So, it is possible to make statements along the line of 'every natural number has such-and-such a property' but you cannot talk about the *set* of natural numbers with some property unless that set is finite. And it is even possible to do mathematical induction, but it has to be done on properties of natural numbers, not *sets* of such.

So, it is possible to define what it means for a natural number to be a prime. It is also possible to show that for every (finite) set of primes, there is a prime that is not in that set. But once again, it is not possible for a finitist to talk about the *set* of prime numbers.

Well, I find this version of mathematics to be, well, very confining. I also find that finitism tends to require rather forced constructions because of its inability to talk about infinite sets.

Now, the circumlocutions and forced constructions seen in finite mathematics are also seen in 'infinitary' mathematics when talking about 'proper classes'. In a real sense, there is an analogy between 'infinite sets' in finitary mathematics and 'proper classes' in set theory: in both cases, we are talking about objects that are 'too big' for the axiom system to deal with. In both cases, there are extensions that allow for at least some discussion of these large objects.

So, it finitism in math worth it? I don't think so. But it is a choice of axiom system as much as anything else. And there are many axioms systems to choose from when you want to play that game.

(April 7, 2019 at 5:25 pm)Smaug Wrote:
(April 7, 2019 at 3:52 pm)Jehanne Wrote: It's a subtle distinction.   Virtually all mathematicians would say that the set of prime numbers is an actual infinite, whose cardinality is identical to the set of natural numbers; most philosophers would agree with this, also.  However, the set of real numbers (both rational and irrational) is, as Cantor proved, an infinite set whose cardinality is greater than the set of natural numbers, even though both are infinite sets.  And, so, some infinities are bigger than others.  The set of natural numbers (and, prime numbers, as well as rational numbers) is a countably infinite set whereas the set of real numbers is uncountable.

What I meant is that if we consider all sets to be fundamentally finite as finitists suggest then there's no need for such a term as cardinality. You can just explicitly specify the number of elements for any given set. And I still can't grasp the finitists' point of view, how they beat the "+1" arguement. For me such recursive definition automatically leads to the notion of infinity. Infinity makes perfect sense from practical point of view, too. In a non-rigorous approach it can be interpreted as a value which is out of scale.

Not quite. When every you do +1 to a finite set, you get another finite set. At no point do you ever construct the *set* of all natural numbers, for example, in finite math.

Standard set theory even has an axiom called the axiom of infinity, which *assumes* there is an infinite set. Without this, all the other axioms would still be consistent. This is why finitism is possible.

(April 7, 2019 at 1:49 pm)Smaug Wrote:
(April 6, 2019 at 10:46 pm)Jehanne Wrote: An actual infinity means a set (say, the natural numbers) has a cardinality that is infinite, as opposed to a "potential infinite", which is always finite, even given the fact such "grows" forever.

Isn't it so that if we invoke the term 'cardinality' we imply that infinite sets exist? Saying that a set can be continued forever is basically saying it's infinite, isn't it?

For the first question, no. Cardinality can be defined perfectly well even when limited to finite sets. The same one-to-one correspondence definition works perfectly well.

The problem in the second question is that we can 'continue forever' but at no point is the set of, say, all natural numbers ever 'reached'.

(April 3, 2019 at 12:10 pm)Jehanne Wrote: I had hoped that Polymath would have responded here, but it appears that he is gone?

I took a short break. I'm back. Smile
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#19
RE: Mathematicians who are finitists.
I find the whole thing to be semantics. If we're dealing with reality, then of course we can have a potentially unbounded and growing value, which is always finite but "potentially infinite". However, in a purely abstract mathematical setting, it’s easy to create infinite sets using particular rules. One can certainly analyse whether the results are internally consistent or not according to those rules, but the set of elements is infinite regardless.

It can be proved that the size of the set is larger than any number, and so it is infinite. The set doesn’t have to "grow" to produce all the elements, they are simply defined into abstract "existence". I suppose you could get into semantics about how exactly the elements are defined, and if that’s done inductively, then you could say the set "never finishes". Well, of course, but this is imposing an artificial limit on a process which doesn’t involve time passing in the first place. You have defined every element in the set, and there are an infinite number of them.

It is fascinating about relative infinities, as has been mentioned with the real and natural (or rational) numbers. I remember being told at uni about how if you compare the density of all the real and rational numbers in any particular interval (say between 0 and 1), the real numbers actually take up all the space. They are infinitely more numerous (thus a higher cardinality). There is no way of placing the real numbers in a list so they can be counted. You will always be missing elements between any list entrants.
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#20
RE: Mathematicians who are finitists.
(April 8, 2019 at 9:56 pm)polymath257 Wrote: As a formalist in my mathematical philosophy, finitism is a possible way to do mathematics. But I find it incredibly limiting.

Here is the basic idea: only finite sets are allowed (actually, hereditarily finite sets). Nonetheless, it is possible to define what it means for a set to be a (Von Neuman) ordinal. For finite sets, this is equivalent to being a natural number. So finitists can talk about natural numbers and finite sets of natural numbers, but there is no way to talk about the set of *all* natural numbers.

So, it is possible to make statements along the line of 'every natural number has such-and-such a property' but you cannot talk about the *set* of natural numbers with some property unless that set is finite. And it is even possible to do mathematical induction, but it has to be done on properties of natural numbers, not *sets* of such.

So, it is possible to define what it means for a natural number to be a prime. It is also possible to show that for every (finite) set of primes, there is a prime that is not in that set. But once again, it is not possible for a finitist to talk about the *set* of prime numbers.

Well, I find this version of mathematics to be, well, very confining. I also find that finitism tends to require rather forced constructions because of its inability to talk about infinite sets.

Now, the circumlocutions and forced constructions seen in finite mathematics are also seen in 'infinitary' mathematics when talking about 'proper classes'. In a real sense, there is an analogy between 'infinite sets' in finitary mathematics and 'proper classes' in set theory: in both cases, we are talking about objects that are 'too big' for the axiom system to deal with. In both cases, there are extensions that allow for at least some discussion of these large objects.

So, it finitism in math worth it? I don't think so. But it is a choice of axiom system as much as anything else. And there are many axioms systems to choose from when you want to play that game.

(April 7, 2019 at 5:25 pm)Smaug Wrote: What I meant is that if we consider all sets to be fundamentally finite as finitists suggest then there's no need for such a term as cardinality. You can just explicitly specify the number of elements for any given set. And I still can't grasp the finitists' point of view, how they beat the "+1" arguement. For me such recursive definition automatically leads to the notion of infinity. Infinity makes perfect sense from practical point of view, too. In a non-rigorous approach it can be interpreted as a value which is out of scale.

Not quite. When every you do +1 to a finite set, you get another finite set. At no point do you ever construct the *set* of all natural numbers, for example, in finite math.

Standard set theory even has an axiom called the axiom of infinity, which *assumes* there is an infinite set. Without this, all the other axioms would still be consistent. This is why finitism is possible.

(April 7, 2019 at 1:49 pm)Smaug Wrote: Isn't it so that if we invoke the term 'cardinality' we imply that infinite sets exist? Saying that a set can be continued forever is basically saying it's infinite, isn't it?

For the first question, no. Cardinality can be defined perfectly well even when limited to finite sets. The same one-to-one correspondence definition works perfectly well.

The problem in the second question is that we can 'continue forever' but at no point is the set of, say, all natural numbers ever 'reached'.

(April 3, 2019 at 12:10 pm)Jehanne Wrote: I had hoped that Polymath would have responded here, but it appears that he is gone?

I took a short break. I'm back. Smile

I'm curious- what sort of insights are gained by using finitist logic?
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