Jump to content
Objectivism Online Forum

Arguments Against Infinite Quantity

Rate this topic


Recommended Posts

I still have the concept of a dragon, fictional or no. If I saw one, I could identify it as one, and I wouldn't protest about calling it a "dragon" because the term only applies to fictional characters.

Why would I need an example of an infinite set? All I need is the notion of infinity in mathematics, then abstracting from sets of empty sets, to the more general setting of sets in general.

And yet again (I think this is the fifth time or more), I am not trying to state a fact of the matter. I am discussing the logic of physical theories and hypotheses, namely this unfounded claim that there can be no finite quantities. I am relatively confident that there is no (strong) proof of an infinite quantity.

Link to comment
Share on other sites

Cantor himself viewed the idea of an infinity of things as contradictory. The reason he did was that, like Aleph's example of perfect squares, a set may be set equal to a proper subset of itself. Instead of that confirming the notion, Cantor took that to show it was illegitimate.

What exists has identity. It must be the same as itself. But some infinities are not the same as other infinities, just in respect of quantity. So an infinity cannot exist. Several proofs of that have, in fact, been put forth in this thread.

Mindy

Link to comment
Share on other sites

Aleph_0 is in this respect of course also a giveaway...

Moreso--it's only natural that I should start the conversation. :)

From what limited grasp of mathematics I have, you cannot apply one to one correspondence to sets that have no definite multiplicity.

Again, I don't know what "no definite multiplicity" could mean. I understand what it is for a set to be not well-defined, but this notion has no meaning to me. If you just mean "not infinite" then that is patently false. The correspondence would be simple: Take any arbitrary natural number, the correspondence associates it to a unique object (in the exact same sense as, for some three objects, there is a correspondence with {1, 2, 3}. What is impossible about this?

Incidently, can this infinite set be used to specifically locate the Dedekind cut for the square root of two?

Since the Dedekind cut for the square root of two is not located anywhere, naturally, no. If you mean, "can we build a correspondence which is a bijection between the real numbers and physical reality?" I don't know, in the same way that I don't know if there is a simple countable infinity. But I don't think it's self-contradictory, so it at least makes sense that we could form a bijection where the Dedekind cut which we identify with the square-root of two being assigned to some physical object. That obviously would not mean that the object would exhibit any behavior or properties that I can imagine, which would be shared by the square-root of two since, obviously, the map is not a homomorphism. I somehow sense this is supposed to be an objection, but I'm grasping at straws about how you might conceive this to be a challenge.

Edited by aleph_0
Link to comment
Share on other sites

Cantor himself viewed the idea of an infinity of things as contradictory. The reason he did was that, like Aleph's example of perfect squares, a set may be set equal to a proper subset of itself. Instead of that confirming the notion, Cantor took that to show it was illegitimate.

Cantor understood that there are "inconsistent multiplicities" (I think that is close to the term he used). But you're comment about sets being equal to a proper subset of themselves is nonsense. By definition, no set is equal to a proper subset of itself. What is the case is that some sets are equinumerous (that is, in 1-1 correspondence with) with proper subsets of themselves (as in Galileo's observation that the natural numbers are in 1-1 correspondence with the squares of natural numbers. And that is not in itself a contradiction and Cantor did not take it as such. The set of natural numbers is equinumerous with a proper subset of itself (indeed, there are many proper subsets of the set of natural numbers that are equinumerous with the set of natural numbers), as we see from Galileo's observation alone, but that is not contradictory. What Cantor referred to as "inconsistent multiplicities" were things such as the set of all sets. But Cantor's work was not axiomatic, and it is in axiomatic set theory that we find that such inconsistent multiplicities are not allowed.

What exists has identity. It must be the same as itself. But some infinities are not the same as other infinities, just in respect of quantity. So an infinity cannot exist. Several proofs of that have, in fact, been put forth in this thread.
One is welcome to make whatever argument against the existence of infinite sets, and there are some mathematicians who reject that there exist infinite sets or who view such existence claims as nonsense. However, what you just wrote is not pertinent to the ordinary treatment of infinite sets in mathematics. In such mathematics, 'infinite' is as an ADJECTIVE. It's not that there is some set that mathematicians call 'infinity' (put aside for a moment locutions such as 'as x goes to infinity', which is another matter and in which 'infinity' drops out in full explication) but rather that certain sets that have the PROPERTY of being infinite. Then, in this mathematics (i.e., standard axiomatic set theory), it turns out that there are there are sets that are infinite but not equinumerous with one another and further that one of the sets has an injection with (a 1-1 correspondence with a subset of) the other, so we say that the second set has greater infinite cardinality than the first set.

Again, I stress that this is aside from whatever other philosophical arguments one might have (whether good ones or not) against the existence or even sense of infinite sets. Merely, that your particular account of the matter is not pertinent as I've shown; you've simply not stated anything that goes on in the ordinary notions in mathematics in which the existence of infinite sets are accepted.

Link to comment
Share on other sites

As of this time, there has been nothing discovered that contradicts the law of identity. To apply it to the universe is a natural extention of the same. When you can validate your claim to the contrary, it will have merit. Proving there is no existential infinity falls into the same arena as proving there is no god.

Link to comment
Share on other sites

From what limited grasp of mathematics I have, you cannot apply one to one correspondence to sets that have no definite multiplicity.

I take it you are talking about mathematics here as standard set theory. In that case, you need to define 'definite multiplicity' in terms of the primitives or previous definitions in set theory.

The notion of a 1-1 correspondence does not require any notion of 'definite multiplicity'. The notion of 1-1 correspondence has a mathematical definition:

S is in 1-1 correspondence with T

if and only if

there exists a function f whose domain is S, whose range is T, and such that for all x and y in S, if f(x) = f(y) then x=y.

If by 'definite multiplicity' you mean that the cardinality operation applies to all sets, then this is proven in ZFC. Every set is in 1-1 correspondence with some ordinal, and we take the cardinality of a set to the least ordinal the set is in 1-1 correspondence.

Note to moderator and others: Again I stress that various philosophical objections may be made to the notion of infinite sets. However, when a critique of the notion is given as to the sense of infinite sets in standard set theory, then that sense should not be misstated nor confused with notions that it does not include.

Link to comment
Share on other sites

As of this time, there has been nothing discovered that contradicts the law of identity. To apply it to the universe is a natural extention of the same. When you can validate your claim to the contrary, it will have merit. Proving there is no existential infinity falls into the same arena as proving there is no god.

A point of logic: how could you "discover" anything that "contradicts the law of identity?" Contradiction depends on identity.

Mindy

Link to comment
Share on other sites

As of this time, there has been nothing discovered that contradicts the law of identity. To apply it to the universe is a natural extention of the same. When you can validate your claim to the contrary, it will have merit. Proving there is no existential infinity falls into the same arena as proving there is no god.

Absolutely nobody has at any point disputed the law of identity. I accept it. The very question is whether infinity is contrary to it, and what reason we have to suppose that it is. The point is to give an argument showing how an injective map as I described above implies the failure of identity, i.e. it is to show that such a mapping implies that there is something which lacks definition, where definition does not just mean "finite or bounded", i.e. the point is to show, in a non-circular way, that the hypothesis of the existence of such a map is self-contradictory. I don't know how else to say it, and I've said it in these ways before many times. Take the hypothesis that there is such an injection, and without assuming that every possible set of disjoint physical objects is finite, without assuming that the law of identity implies finitism, without assuming that "specific" is contrary to having an infinite quantity, provide a contradiction. This may perhaps be done by proving that the law of identity implies finitism, but it cannot be done by assuming it. That is a logical fallacy, called circularity. It may be proved by proving that being a "specific" entity implies being bounded, but it cannot be prove by assuming it, since this is the very thing you hope to prove. I feel like I'm talking to Christians who have been shown their logical fallacy, but without addressing it, continue to employ the fallacious argument, as if they have some need for the argument to be valid, which is more important to them than logic.

Edited by aleph_0
Link to comment
Share on other sites

Note also that if you are to repeat some previous argument to this effect, like the one about boundedness implying no definition, then you need to have some substantial counterargument to my post pointing out that this is an insufficient proof (in that particular case, due to an equivocation of the use of the word "bound"). I can just foresee this kind of thing coming, so as to distract from the lack of any real, working argument.

Link to comment
Share on other sites

Absolutely nobody has at any point disputed the law of identity. I accept it. The very question is whether infinity is contrary to it, and what reason we have to suppose that it is. The point is to give an argument showing how an injective map as I described above implies the failure of identity, i.e. it is to show that such a mapping implies that there is something which lacks definition, where definition does not just mean "finite or bounded", i.e. the point is to show, in a non-circular way, that the hypothesis of the existence of such a map is self-contradictory. I don't know how else to say it, and I've said it in these ways before many times. Take the hypothesis that there is such an injection, and without assuming that every possible set of disjoint physical objects is finite, without assuming that the law of identity implies finitism, without assuming that "specific" is contrary to having an infinite quantity, provide a contradiction. This may perhaps be done by proving that the law of identity implies finitism, but it cannot be done by assuming it. That is a logical fallacy, called circularity. It may be proved by proving that being a "specific" entity implies being bounded, but it cannot be prove by assuming it, since this is the very thing you hope to prove. I feel like I'm talking to Christians who have been shown their logical fallacy, but without addressing it, continue to employ the fallacious argument, as if they have some need for the argument to be valid, which is more important to them than logic.

To rely on the common meaning of terms is not the same as "assuming." When you say, for example, "...without assuming that the law of identity implies finitism..." you seem to exclude direct explanations of why identity is contradicted by the infinite. Cantor saw that the concept was a self-contradiction, because one infinity was not identical with some others, in precisely the respect being attended to.

If you feel as if you're talking to Christians, consider us talking to someone who won't fess up to the limitations of a concept he names himself after--"some need for the argument to be valid?" One might think so.

Mindy

Edited by Mindy
Link to comment
Share on other sites

The universe is not infinite. If it were infinite, there would be no discrete areas within in it. If some set is infinite, you are no closer or further from the end at any point. 1 and 10 are the same, as are 1 and 100,000,000,000,000. This can exist on paper, but it cannot be actualized. There are distinct points in the universe. Otherwise, the phrase "You are here and I am over there." would be semantically and metaphysically null. There is no distinct over here and over there in an infinity.

Link to comment
Share on other sites

First, just because you will be no closer to an end does not mean that there cannot be discrete spaces since there can be an infinity of discrete points. As an analogy, think ofthe integers which stretch to infinity, though you can cordon off finite intervals which contain discrete points.

Second, I'm not convinced that space and material objects are discrete.

However, from the rest of what you wrote, it seems you don't mean "discrete" but "distinguishable". But this seems to assume that the only way to distinguish two objects is by their distance from an end-point. This principle cannot be right, though. Would two objects be indistinguishable simply in virtue of their being the same distance from a third object? Just because they're indistinguishable with respect to one property, doesn't imply that they're indistinguishable simpliciter.

Link to comment
Share on other sites

How is that problematic or relevant? Sure, there are infinitely many transfinite cardinalities. What's the problem?

Link to comment
Share on other sites

Since the thread appears to be devolving into insults at the moment, maybe I'll jump back in and ask some substantive questions.

Aleph_0:

We've established (your point 4 in the OP) that any positive claim of a countably infinite set of physical objects is arbitrary, unless you can provide some sort of very indirect evidence. I want to try to pin you down as to exactly how arbitrary you think it is.

While you seem to accept the higher-level hypothesis that there may exist a scientific theory (as in your post #20) that demands the existence of such a collection, I also take it you are not claiming to have produced an example of such a theory. Such a theory would not simply rely on infinity as a concept of method (such as Newton's infinitesimal division of the Earth to prove gravitational attraction emanates from the Earth's center of mass) but must positively demand an infinite collection of objects. There are two cases.

(1a) Suppose it is shown that no such scientific theory is possible. Is there then any other way in which evidence can be proffered for the existence of an infinite number of physical objects?

(1b) If, for the sake of argument, such a scientific theory was proposed, do you think it would be sufficient to justify the literal existence of such a set of objects?

Next, regarding the acceptance of arbitrary claims:

(2a) If you seriously accept arbitrary claims such as infinite numbers of objects as a possibility due to a lack of self-contradiction, do you regard as equally admissible physical accounts appealing to the existence of Zeus, Ra, etc., as there is equal evidence for both?

(2b) If you object to Zeus, Ra, etc., as being ignoreable by virtue of being unscientific, suppose one hypothesizes a scientific theory mandating the existence of such a god-like being. Would the existence of such a theory in turn demonstrate the existence of our friendly scientific deity? If you reject the possibility of such a theory, in what way is a theory postulating the existence of an infinite collection of objects different in principle?

Link to comment
Share on other sites

You have to like how fast and loose the definition of 'countable' is as well.

To 'count' something, one pairs a unit of what one is counting starting with the first unit in the counting system, and continue pairing the second unit with the second unit from the counting system and so on, until one exhausts the unit's of what one is counting.

At this point it is considered 'counted'.

'Countable' historically has referred to a group of entities of which one can count.

Along comes Cantor. Not satisfied with the limitation imposed by the relation established between the concept and the percieved, sweeps this aside insisting that 'countable' suggest that it can be counted, dismissing that it requires a completion of the task to be considered counted, and states that in an infinite series of counting numbers, they are 'countable', even though it can never be completed. As TheEgoist points out in his post in dealing with infinity, there is no way to determine how far you are from the end at any point.

Galileo made some similar observations about infinity in his observation that dealing with multiple instances of infinity, there is no way to compare equal to, greater than, or less than. He concluded that such comparisons where meaningless.

As Dr. Corvini put it, the propositional method employed by Cantor commits the "logical fallacy of treating an infinite set as if it were a finite group. Treating an open ended classification as if it were a concrete collection."

As Plasmatic points out, he does not prove axiomatic things. The axiomatic is outside the provence of proof. It can be validated, if an individual chooses to validate it for themselves, but an explaination alone does not suffice. The validation, for lack of perhaps a better way of putting it, also has to be 'experiential'.

Edited by dream_weaver
Link to comment
Share on other sites

Join the conversation

You can post now and register later. If you have an account, sign in now to post with your account.

Guest
Reply to this topic...

×   Pasted as rich text.   Paste as plain text instead

  Only 75 emoji are allowed.

×   Your link has been automatically embedded.   Display as a link instead

×   Your previous content has been restored.   Clear editor

×   You cannot paste images directly. Upload or insert images from URL.

Loading...
 Share

  • Recently Browsing   0 members

    • No registered users viewing this page.

×
×
  • Create New...