Axiom of Infinity and Reduction

[aleph_1]

One of the axioms of naive set theory is the Axiom of Infinity (AoI). This implies the existence of the set of Natural Numbers-a set that is actually infinite. One need not object to each individual object within this set to object to the entire set. The existence of this set within set theory has certain phenomenal consequences, such as the existence of many different sizes of infinity. These concepts cannot be reduced to perception and hence are meaningless.

I understand that there are alternatives to naive set theory, such as type theory, but don't know much about that. Does anyone here have an opinion or know of alternatives to naive set theory that allow for mathematics consistent with reduction?

[GrandMinnow]

We should first settle on a definition of 'naive set theory'. Usually naive set theory is understood to have a principle of unrestricted comprehension, so (1) the axiom of infinity is not needed as it is just an instance of unrestricted comprehension, and (2) naive set theory is inconsistent so that perforce no other axioms (such as infinity) are needed.

 

So perhaps it is not naive set theory you have in mind but rather such theories as Z, ZF, ZFC, NBG, etc., which do have the axiom of infinity.

 

As to alternatives, there are many in the literature, including NF, intuitionistic set theories, constructive set theories, even ultra-finitistic set theories (e.g. Levine's 'Understanding The Infinite'), and even more exotic options. However, I cannot say whether any of these meet your criteria of reduction. In any case, it seems fairly clear that there is no formal, explicitly Objectivist-approved axiomatization of the mathematics for the sciences. 

Edited July 29, 2013 by GrandMinnow

[Iudicious]

"These concepts cannot be reduced to perception and hence are meaningless. "

 

Could you clarify what you mean by that? There are plenty of things that cannot be reduced to perception that are chock full of meaning. Just because you cannot "see" infinity per se does not mean it has no meaning. It seems that you're trying to object to the concept of different sizes of infinity, and even to the concept of infinity itself. Correct me if I'm misunderstanding you, because that would be a pretty gigantic leap for someone who has a PhD in mathematics. 

Edited July 30, 2013 by Iudicious

[StrictlyLogical]

I have no PhD.

 

I think "inapplicable (directly) to existents and reality" is the form of "meaningless" used by aleph_1.  These wild abstractions however often ARE useful indirect tools of calculation of quantities which ARE associated with reality.

 

So for abstractions such as the set of Natural Numbers (with an infinite "number" of members) as long as one is aware one is dealing with a pure abstraction all is good.  Although mathematics does require use of concepts such as, infinity, infinite series, limits, etc. math is not reality, it is simply a tool for calculating things which are related to reality.  As such, to reiterate, as long as the distinction between the wild imaginings of Mathematicians and reality are kept clear, there is no problem with abstractions like the Natural numbers.  As an aside I think it using words such as "actual", and "exists" are problematic to use in mathematics... but this is convention.

 

 

As an aside:

 

Q:  What is more correct regarding a mathematician's view of the concept of "infinity" is it a "quantity" or a number like 10, or is it indicative like a direction e.g. "North".

 

 

In some sense I can understand infinity as not meaningless if I think of it as a direction:  Q: where are you? A: 10 feet North of x.  Q: where are you heading?  A: TOWARDS North, the direction.

[GrandMinnow]

What is more correct regarding a mathematician's view of the concept of "infinity" is it a "quantity" or a number like 10, or is it indicative like a direction e.g. "North".

Let's set aside for the moment such things as "points of infinity" in the extended real number system. Instead, let's look at the notion of infinity as mentioned in this thread, specifically the infinitude of the set of natural numbers. In this sense, it is more accurate to refer to the adjective 'is infinite' than to 'infinity'. 

 

That is, in set theoretic mathematics, we have the property 'is infinite'. There are two prominent defintions, of which one may choose to use either:

 

(1) S is infinite if and only if no natural number is in 1-1 correspondence with S.

 

(2) S is infinite if and only if S is in 1-1 correspondence with some proper subset of S.

 

In Z set theory, we have that the definiens of (2) implies the definiens of (1). That is, Z proves that if S is in 1-1 correspondence with a proper subset of itself then S is not in 1-1 correspondence with any natural number. (This is the famous "pigeonhole principle".)

 

But not even ZF set theory proves that the definiens of (1) implies the definiens of (2). That is, ZF does not prove that if S is not in 1-1 correspondence with any natural number then S is in 1-1 correspondence with some proper subset of S.

 

To prove that the definiens of (1) implies the definiens of (2), ordinarily some form of a choice principle is used, such as the axiom of choice or the weaker axiom of countable choice. 

 

Terminologically, "S is in 1-1 correspondence with a proper subset of S" is said as, "S is Dedekind infinite" while "S is not in 1-1 correspondence with any natural number" is said as "S is infinite" or sometimes "S is Tarski infinite". 

Edited July 30, 2013 by GrandMinnow

[StrictlyLogical]

I take it S are sets.

 

 

Are these S's those sets of sets of...(etc.) empty sets?

 

 

What does 1-1 "correspondence" mean when corresponding and entirety of membership of a set with a "proper subset" of that same set?

 

i.e. how can a set S BOTH have "1-1 correspondence" with itself, i.e. its entirety (I assume it does) AND have "1-1 correspondence" with a "proper subset" of S. If S is not identical with its "proper subset", what is that which is "left over" i.e. a member of S but not a member of its proper subset?  How "big" is that?  Does it have any "type" of "correspondence" to anything?

 

Alternatively:  IF we define "1-1 correspondence" as that which a set S (we happen to call infinite) has with itself and also with a "proper subset" of S... what HAVE we included in the definition of 1-1 correspondence?

[aleph_1]

i.e. how can a set S BOTH have "1-1 correspondence" with itself, i.e. its entirety (I assume it does) AND have "1-1 correspondence" with a "proper subset" of S.

You need to look up Hilbert's Hotel.

[aleph_1]

GrandMinnow,

My problem relates to the analytic-synthetic dichotomy as well. Some here seem willing to take mathematical concepts as abstractions disconnected from the synthetic. Splitting infinities is the ultimate split in this regard. We mathematicians sling around alephs without regard to any synthetic correspondence. The ideas that lead to this split are my interest. Thank you for your input. You have given me something to mull over.

[aleph_1]

"These concepts cannot be reduced to perception and hence are meaningless. "

Could you clarify what you mean by that? There are plenty of things that cannot be reduced to perception that are chock full of meaning. Just because you cannot "see" infinity per se does not mean it has no meaning. It seems that you're trying to object to the concept of different sizes of infinity, and even to the concept of infinity itself. Correct me if I'm misunderstanding you, because that would be a pretty gigantic leap for someone who has a PhD in mathematics.

Reduction is the key to avoiding the analytic-synthetic dichotomy. Oist philosophy takes the meaning of a first-order concept to be its existents. Higher-order concepts must be reducible to lower order, terminating with existents. What two collection of existents can I find where one is countably infinite and the other is uncountably infinite?

[GrandMinnow]

I take it S are sets.

Yes.

 

Are these S's those sets of sets of...(etc.) empty sets?

Any set. Of course, the empty set in particular is finite.

 

What does 1-1 "correspondence" mean when corresponding and entirety of membership of a set with a "proper subset" of that same set?

A 1-1 correspondence between S and T is a bijection from S onto T. It is not precluded that T may be a proper subset of S.

 

S and T are in 1-1 correspondence if and only if there is 1-1 correspondence between S and T. 

 

That there are sets in 1-1 correspondence with proper subsets of themselves is easy to show by such as exmaples as this:

 

Let 'w' stand for the set of natural numbers (i.e., the set whose members are 0, 1, 2, ...). Let f be the function whose domain is w and such that for all n in w, we have f(n) = n+1. So f is a 1-1 correspondence between w and w\{0}, i.e., between the set of natural numbers and the set of positive natural numbers. 

 

Another example, let f be the function whose domain is w and such that for all n in w, we have f(n) = 2*n. Then f is a 1-1 correspondence between the set of natural numbers and the set of even numbers.

 

Examples abound ...

Edited July 30, 2013 by GrandMinnow

[StrictlyLogical]

I guess I was assuming the definition was going to "generate" the concept of "is infinite".  i.e. start from that which is not "already" infinite... and define how to get there. 

 

It seems that we simply hold up the already "finite" or "infinite" abstraction up to the standard test and determine its "status".  all x in N and all y = 2k where k in N are both countably infinite.

 

 

It's been years since I took set theory.  I seem to recall something about integers being represented by sets of the empty set, or sets of sets or power sets of them... anyway, since university I've always understood the abstraction infinity or infinite but since learning of Objectivism I have not been able to reduce the concept to existents themselves, only to other abstractions (like length or volume of "space") in relation to existents, or contingent-hypotheticals (not sure of the philosophic jargon) such as "I can throw the ball in an infinite number of directions" (whereas afterwards I threw it in only one direction). 

 

 

Is plain old countable infinity reducible to anything(s) in actual existential reality (and not an abstract projection onto it)?

 

 

If so uncountable infinities and the higher order Alephs (is that right?) are definitely out there...

[GrandMinnow]

I guess I was assuming the definition was going to "generate" the concept of "is infinite".  i.e. start from that which is not "already" infinite... and define how to get there.

The definition of 'is infinite' is as I stated it. In sense (1) I mentioned, it's simply the negation of finitude: S is infinite if and only if S is not finite. It's not given through a notion of "getting there". However, there is a set theoretic notion that you might capture what you have in mind by "getting there", but that notion does not itself play a role in defining 'is infinite'.  

 

It seems that we simply hold up the already "finite" or "infinite" abstraction up to the standard test and determine its "status".

If I understand what you're getting at then yes, that is the idea. We have a definition of 'finite' and for any given S, either S meets the criteria of that definition or it does not (same for 'infinite'). 

 

all x in N and all y = 2k where k in N are both countably infinite

If I understand you correctly, you're saying this [where 'N' stands for the set of natural numbers]:

 

{x | x in N} is countably infinite; and {y | there is a k in N such that y=2k} is countably infinite. I.e. both the set of natual numbers and the set of even numbers are countably infinite.

 

And that is a correct statement.

 

Is plain old countable infinity reducible to anything(s) in actual existential reality (and not an abstract projection onto it)?

I don't know, and I'm not committed to an Objectivist notion of reality such as I surmise you have in mind.

 

If so uncountable infinities and the higher order Alephs (is that right?) are definitely out there...

I don't know what "out there" means. Whether Objectivist or not, I think we can agree that these mathematical sets are abstractions and not material objects. In what sense one wishes to say that such abstractions exist or even make sense is up to one's philosophy or framework. Objectivism rejects that there exists any "complete infinity" but accepts a notion of "potential infinity". Whether that entails that Objectivism rejects even the existence of infinite sets in the mathematical sense of "exist" as in such mathematical statements as "there exist sets that are infinite, i.e. there are sets that are examples of "complete infinities"" (commonly accepted by mathematicians) is up to Objectivists to say.

 

However, if one withholds adopting the power set axiom, then it would not be inconsistent to hold that there exist countably infinite sets but that there do not exist uncountable sets. But, again, whether Objectivism accepts that such things as the power set axiom are even meaningful is up to Objectivists to say.

 

Meanwhile, ordinary set theoretical mathematics does not use a notion of "potential infinity" and it's not apparent even how that notion could be formally incorporated into ordinary set theoretical mathematics. 

Edited July 31, 2013 by GrandMinnow

[GrandMinnow]

higher order Alephs (is that right?)

Maybe some authors use the term 'higher order alephs', but I would just say 'higher alephs' or 'uncountable alephs'.

 

aleph_0 is the least aleph and it is the only countable aleph.

 

For any ordinal k>0, we have that aleph_k is a "higher aleph" or "uncountable aleph".

[StrictlyLogical]

 

 

Maybe some authors use the term 'higher order alephs', but I would just say 'higher alephs' or 'uncountable alephs'.

 

aleph_0 is the least aleph and it is the only countable aleph.

 

For any ordinal k>0, we have that aleph_k is a "higher aleph" or "uncountable aleph".

 

Do higher alephs have any indirect practical application?  Can they be used to prove or calculate things which are useful in realms of math which can relate to reality even remotely?

[GrandMinnow]

Do higher alephs have any indirect practical application?  Can they be used to prove or calculate things which are useful in realms of math which can relate to reality even remotely?

I don't know. 

 

However, consider that the set of real numbers is an uncountable set. And the real numbers and the calculus of the real numbers is the starting context for mathematics for the physical sciences. But it turns out that while the set of real numbers is uncountable, our set theory does not determine what aleph is the cardinality of the set of real numbers (this is the famous continuum problem). In other words, the theory proves that the cardinality of the set of real numbers is an aleph, but the theory does not prove which aleph it is. 

 

Also, consider the context here. Suppose we wish to have an axiom system (I don't mean 'axiom' in the Objectivist sense) from which we can prove the various theorems that are used in mathematics for the physical sciences. And suppose we wish for the mechanics and fine points of this axiom system to be as easy to use and comprehend as possible. Well, many people feel that modern axiomatic set theory is such a system. However, suppose these axioms provide MORE than needed for the physical sciences, as the axioms provide for such things as higher and higher infinite cardinalities without end. And suppose there does not seem to be a practical way to cut the axioms down so that they give us only scientific mathematics but not higher and higher infinite cardinals. Well, that is pretty much the situation we're in.

 

ZFC is pretty much standard modern set theory; however the somewhat weaker (Z\"regularity")+"dependent choice" (which I'll call "ZS") is considered adequate for the physical sciences. Now ZS is as described in the above paragraph. It gives us mathematics for the sciences but it overshoots by giving us greater and greater infinite sets. This set theory is like a strange high jumper who can always clear the bar at 7 feet but he can only do it by clearing 8 feet (he can't help himself from jumping at least 8 feet if he's required to jump at least 7 feet). 

 

Now, one can propose a system that does not overshoot. But how to do it and keep the complexity of working in the system manageable? Of course, 'manageable' is subjective, so it's up to each person to decide for himself whether any of the proposed alternatives are manageable.

Edited July 31, 2013 by GrandMinnow

[GrandMinnow]

Also just as important to mathematicians is that the axioms are in keeping with our basic notions of sets. Consider the axioms of full ZFC [all the variables mentioned here range over sets]:

 

(1) Extensionality. Sets are determined entirely by their members. That is, X and Y are equal if and only if every member of X is a member of Y and vice versa.

 

(2) Union. For any X, there is the union of X. That is, for any X there is the set whose members are exactly those that are a member of some member of X.

 

(3) Power set. For any X, there is the power set of X. That is, for any X there is the set whose members are exactly those that are subsets of X.

 

(4) Schema of Replacement. For any X and any functional property from X, there is the set whose members are exactly those that are the correlated set by the functional property of some member of X. That is, for any X and any correlation from X, there is the set whose members are exactly the sets that are correlated to.

 

(5) Infinity. There is a set that has the empty set in it and has in it the successor of any member. (The empty set will have been defined and proven to exist by axioms (4) and (1). And the unordered pair operation and singleton operations while have been defined from axioms (3) and (4). And binary union will have been defined by axiom (2) and the unordered pair operation. So that 'successor' is defined by 'the successor of n = the binary union of n with the singleton whose only member is n'.) 

 

(6) Choice. For any X that does not have the empty set has a member, there is a set "made" by "collecting" from X exactly one member of each member of X. That is, there is a choice function for every set that does not have the empty set as a member. 

 

(7) Regularity. For any nonempty X, there is a member y of X such that no member of X is a member of y. That is, every set has a membership-minimal element. This entails that sets are not members of themselves and don't have infinitely descending membership sequences, etc. 

 

/

 

It's a pretty safe bet that the vast majority (there are some dissenters) of mathematicians (who are even concerned about such foundational matters) find these principles to be faithful to our notion of sets.

Edited July 31, 2013 by GrandMinnow

[aleph_1]

I have to relate a remarkable story. A ten-year-old who took Calculus 3 from me came to my office and asked whether there was an aleph-one-half. I ascertained that he understood what he was talking about by having him prove that the real numbers are not countable on my board, and then showed him a reference discussing the continuum hypothesis. (Standard math assumes there is not but there is if you want there to be.) Is this not pertinent to the present discussion?

Playing with infinities may be fun but isn't it ultimately about what you want or are willing to accept? When the nature of infinities depends only on your whim, does that not mean that they are detached from reality and hence meaningless (in the Oist sense)?

By the way, following what the majority does is a recognized form of fallacy. Just because the VAST majority of mathematicians accept something does not make it sacrosanct.

Edited July 31, 2013 by aleph_1

[aleph_1]

Just in case you are curious, that ten-year-old went on to take linear algebra and differential equations from me, has now earned his 4year degree and moved on to graduate school. I believe that he is now 14.

[aleph_1]

GrandMinnow's point that standard math provides more than is needed for the physical sciences is an interesting one. The point amounts to the idea that one's abstractions may be richer than reality. Some argue that this is necessary and beneficial. My understanding of Oism is that this is inconsistent with Oism.

[GrandMinnow]

delete

Edited July 31, 2013 by GrandMinnow

[GrandMinnow]

asked whether there was an aleph-one-half. I ascertained that he understood what he was talking about by having him prove that the real numbers are not countable

The answer to that question is no, by definition, there is no aleph_one-half. Asking whether there is an aleph_one-half shows a very basic lack of understanding of what the alephs are. Showing the uncountability of the reals has nothing to with it.

 

When the nature of infinities depends only on your whim, does that not mean that they are detached from reality and hence meaningless (in the Oist sense)?

(1) It can be argued that the axioms are not merely whimsical and that, as I just mentioned in my previous post, the axioms reflect our (editorial 'our') basic notion of "set". (2) Moreover, the axiomatic method does not need to be understood as making flat unqualified assertions but rather as providing a set of assertions that are true only of certain structures. (3) As I've been saying, it's not clear that Objectivism endorses even the notion of formal axiomatics. If one does not seek a formal axiomatization of the mathematics for the sciences then one does not even have any skin in the game of trying to get axioms that provide the mathematics for the sciences while not overshooting. Back to the high jumper analogy regarding this.

 

By the way, following what the majority does is a recognized form of fallacy. Just because the VAST majority of mathematicians accept something does not make it sacrosanct.

And of course I'm not arguing that anything is settled by majority opinion. Rather, as you can see from the context of my posts, I'm only pointing out the fact that the goal of an axiomatization of the mathematics for the sciences is acheived by set theory faithful to mathematicians' notion of "set". I'm not arguing that that mere consensus makes set theory otherwise "meaningful", correct, or whatever. 

Edited July 31, 2013 by GrandMinnow

[aleph_1]

The generalized continuum hypothesis is that if n is a cardinal number, then there is no cardinal number between n and 2^n. We ordinarily take aleph1 to be 2^aleph0. It is a mere hypothesis that there is no cardinal number between. You may, at your whim, take such an intermediate to be aleph one-half. Okay, that may be an abuse of the aleph notation, but you might call it k and the concept he was getting at still makes some sense. I do not think that this is a misunderstanding. I also do not think it inappropriate to ask a kid to demonstrate some understanding of cardinality before taking his question seriously. He demonstrated to me knowledge of countable infinity and also a proof that the real numbers are not countably infinite.

Showing the uncountability of the reals at minimum shows that there are distinct notions of infinity. My claim is that theories that split infinities contain meaninglessness. Is this not relevant?

Note: That aleph1=2^aleph0 may be taken as the continuum hypothesis. Is this not so?

Edited July 31, 2013 by aleph_1

[GrandMinnow]

"The generlized continuum hypothesis is that if in is a cardinal number, then there is no cardinal number between n and 2^n." 

 

Yes, where n is infinite.

 

"We ordinarily take aleph1 to be 2^aleph0."

 

No, we do not. To say that aleph_1 = 2^(aleph_0) is to adopt the continuum hypothesis. But the continuum hypotheis is independent of ZFC. Without adopting the continuum hypothesis, ZFC is agnostic as to the question whether aleph_1 = 2^(aleph_0).

 

"It is a mere hypothesis that there is no cardinal number between."

 

All the axioms are hypotheses. And ZFC does NOT include the continuum hypothesis as an axiom.

 

"You may, at your whim, take such an intermediate to be aleph one-half."

 

No, you may not COHERENTLY do so. The alephs are by DEFINITION an operation on the ordinals. To mention an aleph subscripted by anything other than an ordinal is nonsense and indicates that the person does not know what the alephs are. If one wishes to define some OTHER operation that takes arguments that include non-ordinals then fine, one can go ahead and state the definition, but then it is something other than the aleph operation. 

 

"I do not think that this is a misunderstanding."

 

It is a FUNDAMENTAL misunderstanding.

 

"I also do not think it inappropriate to ask a kid to demonstrate some understanding of cardinality before taking his question seriously."

 

I don't think it's inappropriate to ask him anything about mathematics. I'm just pointing out that, whether the person is ten days old or ten millennia old, the uncountabilty of the reals does not bear upon explaining why the expression "aleph_(one-half)" is a fundamental confusion.

 

"Showing the uncountability of the reals at minimum shows that there are distinct notions of infinity."

 

Not different notions of 'is infinite' but rather that there are infinite sets of different cardinality. Anyway, it doesn't take the example of the reals to achieve this result. More simply: By an easy proof, every set is dominated by its power set. Therefore, the power set of the set of natural numbers is uncountable.

 

"My claim is that theories that split infinities contain meaninglessness. Is this not relevant?"

 

I don't begrudge you from holding that set theory is meaningless. I'm just pointing out that the answer to "is there an aleph_(one-half)?" is no, there is not, by the very definition of the aleph operation, and I'm pointing out that the uncountability of the reals (and now in the discussion) the continuum hypothesis is extraneous to the explanation.

Edited July 31, 2013 by GrandMinnow

[GrandMinnow]

By the way, my all-caps are not meant as shouting, but rather as tantamount to italics (without using special coding).

Edited July 31, 2013 by GrandMinnow

[GrandMinnow]

If I understand you correctly, your idea seems to be that without the continuum hypothesis there may be a cardinality between aleph_0 and aleph_1, which could be called "aleph_(1/2)"? Is that your idea? If so, let me explain why it is confused:

 

(1) The aleph operation is defined by transfinite recursion on the ordinals. The alephs do not take arguments other than ordinals. Period.

 

(2) Without the continuum hypothesis there may be a cardinality between aleph_0 and 2^(aleph_0). But without the continuum hypothesis there is still no cardinality between aleph_0 and aleph_1. By DEFINITION, aleph_1 is the least cardinal greater than aleph_0. By DEFINITION, irrespective of the continuum hypothesis, there is no cardinality between aleph_0 and aleph_1. 

 

Put another way, if we negate the continuum hypothesis, then still there is no cardinality between aleph_0 and aleph_1 (by DEFINITION, irrespective of the continuum hypothesis, aleph_1 is the least cardinal greater than aleph_0); rather, it's just that when we negate the continuum hypothesis it is not the case that aleph_1=2^(aleph_0).  

 

P.S. ZFC does not include the continuum hypothesis, and ZFC does not prove one way or the whether aleph_1=2^(aleph_0). But still ZFC does prove that there is no cardinality between aleph_0 and aleph_1, as that is true by DEFINITION.

Edited July 31, 2013 by GrandMinnow