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are they infinite numbers ?

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are they infinite numbers ?
eldorado
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Posted 07/23/09 - 12:11 AM:
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#11
swstephe wrote:
Don't think of numbers and individual abstract symbols, (like a vocabulary), but a system of describing quantities, (like a grammar). The grammar of producing numerical values, (like natural numbers), is potentially infinite, (although there are some problems with that on the very fringe). But remember, "infinite" really means "no limit", it isn't a specific value. Your brain can contain an infinite set represented by a grammar to produce that set.


Do You mean that when I say "infinite set of numbers" I only mean that i know the rule to produce any number belongs to this set ?

in other words , infinite set of numbers means only the rule which used to produce the numbers not existance of infinite quantity of numbers in my brain ?

thanks alot for your help.
AKG
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Posted 07/23/09 - 12:14 AM:
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#12
... or more succinctly: the axiom of infinity is a sentence in the formal, first-order language of set theory, and it asserts that yes indeed, there exists an infinite set (it asserts something stronger in fact, but never mind that). You disagree with this axiom because:

a) you don't think the existence of infinite sets has anything to do with the axiom of infinity.

b) you think that if there were a correct way to express the existence of an infinite set, the axiom of infinity would be it, but you think that no mathematical objects exist so the axiom of infinity ends up being a purely syntactic expression with no semantic content

c) you think it's meaningful, you think it's in a language that talks about mathematical objects which really exist, but you think that this specific axiom is false
"The only reason we die... is because we accept it as an inevitability." -- Stewie

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MoeBlee
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Posted 07/23/09 - 11:21 AM:
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#13
AKG wrote:
it's good to see some old names - MoeBlee et al.
Nice to see you too, AKG.
swstephe
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Posted 07/23/09 - 06:39 PM:
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#14
eldorado wrote:
Do You mean that when I say "infinite set of numbers" I only mean that i know the rule to produce any number belongs to this set ?

in other words , infinite set of numbers means only the rule which used to produce the numbers not existance of infinite quantity of numbers in my brain ?


When you say "infinite", you are really saying, "without limit or bound". That is simply an unbounded set whose members are defined by a constraint. There are no limits to the numbers you could conceivably produce from a sequence of digits. That is quite different than saying "there is a bag holding every number I could ever produce". There are some transformations you can do on sets to show this. The set of all positive integers has no bound, by definition. Take every other number, starting with "0", and you have the set of all even positive integers, it also has no bound -- the size of both sets is the same even though you are selecting only half of the members. This shows that the cardinality doesn't change when you try to restrict the grammar that generates the members of the set.
Ethics is the measuring of morality. Morality is the measuring of good. Good is the measuring of benefit. Benefit is the measure of values.
muxol
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Posted 07/23/09 - 08:57 PM:
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#15
AKG wrote:
despinozist, you say you don't believe in infinite sets; is it for one of the following reasons:

a) you reject all of modern mathematics altogether. Nearly every branch of mathematics invokes the idea of infinite sets, and moreover infinite sets of various sizes (some are bigger than others).


What notion of "infinite set" do you have in mind? Intuitionists, for example, reject the classical notion of infinite set as a completed infinity. In this sense they reject the existence of infinite sets, *in the classicists sense*, and this is the sense employed by most concerned with foundational issues in mathematics. (Outside of that group of people, working mathematicians without any interest in foundational matters don't really employ the concept SET (using capitalized expressions to denote concepts), or at least don't need to. E.g. it is a highly contentious claim in the philosophy of mathematics that real analysis involves essentially the concept SET. But more generally, category theory give set theory a run for its money in terms of the foundations of math, and one could easily make the case that CATEGORIES are more fundamental a notion in mathematics than are sets (and membership, etc.)

b) You accept modern mathematics the way it is, but you're an anti-platonist, and thus regard it as a big game of consistent symbolic manipulation. In this case infinite sets don't exist, but neither do finite sets. Indeed in this case no mathematical objects exist, they're all just figments of our imaginations used to try to give some semantic/intuitive content to the purely syntactic manipulations that mathematics really is.

c) You accept modern mathematics and are a platonist, but somehow reject the axiom of infinity.


One other very plausbile option is that he's a Platonist AND rejects the notion of SET (e.g. as coherent) or that there are such things as sets (but might accept that the notion is coherent but not tied to anything actual). I can believe in the real numbers *without* believing in sets. Generally, I can believe in pluralities without believing in sets (cf. Boolos on plural quantification).

In any case, what did the OP mean by 'infinite numbers'? Was he committed to the reduction/identification of numbers to sets of a certain kind? I don't know, because I didn't read the rest of the thread, but it is at least consistent to reject that reduction, and to reject infinite sets while remaining a platonist about mathematical (barring set-theoretic) discourse.
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Posted 07/23/09 - 09:38 PM:
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#16
muxol wrote:
What notion of "infinite set" do you have in mind?
A set from which there is no injection into any natural number.
Intuitionists, for example, reject the classical notion of infinite set as a completed infinity.
What's the difference between a "completed" infinity and an "incomplete" infinity, and can this difference be stated in mathematical terms? Does rejecting "completed infinites" mean rejecting the axiom of infinity, or is the axiom of infinity not even expressed in the right language to talk about whether or not infinite sets exist.
In this sense they reject the existence of infinite sets, *in the classicists sense*, and this is the sense employed by most concerned with foundational issues in mathematics. (Outside of that group of people, working mathematicians without any interest in foundational matters don't really employ the concept SET
What exactly do you mean by that? I'm not an algebraist or an analyst, but I've studied these things and they seem to use the concept of sets in the same way that foundationalists do, it's just that they don't study sets in general. But they study their specific sets, and they treat them like sets the same way set theorists treat sets, or so it would seem to me.
(using capitalized expressions to denote concepts), or at least don't need to. E.g. it is a highly contentious claim in the philosophy of mathematics that real analysis involves essentially the concept SET. But more generally, category theory give set theory a run for its money in terms of the foundations of math, and one could easily make the case that CATEGORIES are more fundamental a notion in mathematics than are sets (and membership, etc.)
I don't think that's a very commonly held view, and I think much of mathematics already uses the framework and semantics of set theory, I can't see why one would want to switch to category theory. Even in the category theory I've studied, a concerted effort is made to reconcile the contradiction between set theory and say a "naive" approach to category, since things like the category of all groups doesn't exist as a set. I'm interested though, could you elaborate on your point?
One other very plausbile option is that he's a Platonist AND rejects the notion of SET
I suppose it's possible, but why would one do that?
"The only reason we die... is because we accept it as an inevitability." -- Stewie

"To enslave nuance to dogma is folly." -- Lord Hillyer
eldorado
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Posted 07/24/09 - 06:05 AM:
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#17
swstephe wrote:


When you say "infinite", you are really saying, "without limit or bound". That is simply an unbounded set whose members are defined by a constraint. There are no limits to the numbers you could conceivably produce from a sequence of digits. That is quite different than saying "there is a bag holding every number I could ever produce". There are some transformations you can do on sets to show this. The set of all positive integers has no bound, by definition. Take every other number, starting with "0", and you have the set of all even positive integers, it also has no bound -- the size of both sets is the same even though you are selecting only half of the members. This shows that the cardinality doesn't change when you try to restrict the grammar that generates the members of the set.


Thanks alot.

So, the infinite set is just abstract possibility to produce any quantity of numbers and is not existance of infinite quantity of numbers.

If I understand this well , how could be some sets bigger than others ?

Set of rational numbers is bigger than the set of natural numbers. So , is the meaning of that when we take any interval like [1,5] , the rational numbers we can produce is more than natural numbers ? or there is different meaning.

Thank you very much for help.
MoeBlee
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Posted 07/24/09 - 09:25 AM:
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#18
eldorado wrote:
Set of rational numbers is bigger than the set of natural numbers.
Under what definition of 'bigger'?
eldorado
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Posted 07/24/09 - 12:04 PM:
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#19
MoeBlee wrote:
Under what definition of 'bigger'?



That is what I ask about.
MoeBlee
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Posted 07/24/09 - 12:21 PM:
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#20
eldorado wrote:
That is what I ask about.
In ordinary set theory, we have these definitions:

y dominates x if and only if there is a 1-1 function from x into y

y and x are equinumerous if and only there is a 1-1 function from x onto y

y strictly dominates x if and only if y dominates x and y and x are not equinumerous.

So you can think of 'y dominates x' as 'y is bigger than or the same size as x'; 'y strictly dominates x' as 'y is bigger than x'; and 'y and x are equinumerous' as 'y and x are the same size'.

With those definitions, the set of natural numbers is the same size as the set of rational numbers, since there is a 1-1 function from the set of natural numbers onto the set of rational numbers.
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