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Axiom Of Choice?

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David emphasized the word "generated".

In one sense of the term, everything on your computer screen is generated by the computer. In another sense of the term, it is all generated by the manufacturer and programmers. So, when you say the following:

it appears that you are either equivocating on the word "generated", or using it in a specific sense which needs to be made clear.

I understood his emphasis, but I think his problem was really with the term "meaning" (I use quotation marks because I don't have access to italics). I think he thought "generating meaning" was an odd usage (indeed, I think the world meaning here admits of too many borderline cases and has too wide a usage for me to make myself perfectly clear using it). I think the idea of generating words is pretty self explanatory. If he, or you, have a problem with my usage I will need you to elaborate your question since generate is a pretty basic concept.

(Fixed quote tags - softwareNerd)

Edited by softwareNerd
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It sounds like he was.

No offense, but I think my ideas my have been difficult for Ashley to articulate. Please don't accuse me of being "mistaken or lying". Arguments in foundations of mathematics require a bit of technical expertise, an expertise quite like Ashley's in Biology---a subject where all I have is slightly above basic college level attainment.

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I understood his emphasis, but I think his problem was really with the term "meaning" (I use quotation marks because I don't have access to italics).
First off, you do: alt-I toggles italics (also: it's a good idea to preview, so you don't mess up the pairing of tags). Second, my problem was with the combinbation. "Generating words" is not self-explanatory. I doubt you mean "provide an explicit formal account of", to point to one use of the term. Does a hammer generate a house, or does a carpenter generate a house using a hammer? More closely analogous, does a stero generate music, or does a stereo manifest music which was generated by a composer and / or musician? I'm just trying to make sense of the idea of things having meaning without a meaner. I understand what it means for a person to try to infer a meaning interpretation to a string of words, based on what they think they might have meant if they had uttered such a string. But then that is a sensible thing to do only if you think the string represents something with meaning. This boat eat word but dog throw rough a salad for blue across ribbon know you cry car hammer random with the all under through if some big while is. I just don't get the idea of detached meaning without a meaner.
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First off, you do: alt-I toggles italics (also: it's a good idea to preview, so you don't mess up the pairing of tags). Second, my problem was with the combinbation. "Generating words" is not self-explanatory. I doubt you mean "provide an explicit formal account of", to point to one use of the term. Does a hammer generate a house, or does a carpenter generate a house using a hammer? More closely analogous, does a stero generate music, or does a stereo manifest music which was generated by a composer and / or musician? I'm just trying to make sense of the idea of things having meaning without a meaner. I understand what it means for a person to try to infer a meaning interpretation to a string of words, based on what they think they might have meant if they had uttered such a string. But then that is a sensible thing to do only if you think the string represents something with meaning. This boat eat word but dog throw rough a salad for blue across ribbon know you cry car hammer random with the all under through if some big while is. I just don't get the idea of detached meaning without a meaner.

Meaning is seperate from intent. Also, I think there is a tendency in Peikoff's work on logic to draw distinctions that are more annoying than helpful. Whenever a concept is dileated, it is dileneated with respect to a certain telos: It has an applicability. Why does he introduce the meaning must have consciousness distinction? Why is it useful to say that a computer program can't therefore have meaning? [My answer is that he wanted to combat nominalism, but he went a little too far and failed to realize what the full implications of his formalization would be. Namely, a destruction of mechanized logic]

In fact, I think that computers provide a case and point against his over simplified classification. They clearly generate meaning (I apologize for using this term again) even though they do not have consciousness. You can see what they print out and determine---as long as nothing mechanical went wrong---makes reference to something even though the computer does not understand the meaning.

When I say that computers generate words that have meaning, I am saying that they follow---much as people do---a set of rules by which the words they put up on their screens make reference to some state of affairs in the world. The computer can only make statements about the data it is feed or its internal state (however, this is true for people as well), but they have meaning because they are subject to a set of strict criteria that determine what words are used under what circumstances and that insure they have references to real objects with real qualities and dimension. This is how they have meaning, and----since we cannnot know the consciousness of another person---this is how we determine what other people mean as well. The reason the parrot cannot create meaning is that it does not consistently speak with respect to a strict grammar. Since the grammar and laws of reference are not the cause of the bird's speech, it does not have meaning.

To give an example of why this way of thinking about meaning is more precise, consider this popular use of the term meaning: It didn't rain all summer: that means our crops are likely to fail. I think your rural farmer is onto something here. The test of meaning is reference to an external state of affairs and to causality: the speech (or sign or symbol) has to reference some verifiable state of affairs and the words have to be consistently "caused" (but not necessarily cause, see Mill's Methods on determining causality) by that external state. Because consciousness is hard/impossible to quantify and frequently even to describe, it is unwise to create theories that make excessive reference to this faculty. It should only be brought into the equation when it is most necessary.

Now, please don't assume that I think consciousness is not a given, or that I think it should never be referenced: It is an aspect of reality and, therefore, must form a significant part of many theories about reality. I simply wish to apply Occam's razor. Why reference the consciousness every use of the language two people when you can create a theory that only references the consciousness of those who determine the proper use of language.

I, for example, do not think there is evidence that animals possess consciousness in anything like our meaning of the term, but I do think that some animals may have rudimentary forms of communication. How can you account for this by your theory? When a monkey screams because a lion is around, I can here that scream and think "I must get the fuck out of here", yet the monkey possesses no consciousness. Peikoff's theory would have to grant animals "consciousness", which is patently absurd.

Here is why, lastly, specific consciousness should be referenced as little as possible when we use the term meaning. What if you say, " I read Anthem." The teacher than replies "But the Fountainhead was the assigned book." Oh, I intended to say "Fountainhead". Would you then claim that your sentence had a different meaning from the teacher took it to have simply because your intent was different. This leads language into a paultry sort of subjectivism, and this is why I think the role of consciousness should only be applied to the creation of the categories and not to any specific application.

Consciousness plays its key role in the establishment of the categories of language and the development of conception, but once these rules are established we quickly fall down a slippery slope if we insist that a conscious integration on the part of the speaker is required to determine meaning.

Lastly, meaning should be considered seperate from intent. Meaning is the by product of the conscious integration and categories developed at the outset (or agreed upon during the development of) a language. Intent is the goal of the speaker. I will readily say that computer's don't have intent, but the words have meaning because (as I said before) their sentences describe the world.

I will provide one last example. Stoplights are often programmed to randomly switch between go and stop. Now, we clearly accept that this has meaning, but the meaner was the people who established the convention of red, green, and yellow.

My final point, and I apologize for not structuring this better, is that Peikoff is right to say that consciousness must play some role. But it plays its role in establishing the rules and categories. After that, it is no longer a direct byproduct of the consciousness of its particular user. To try to put this more simply. The role of consciouness is the establish the rules and formulation. After that, meaning can be determined by analyzing the form in context.

Edited by Franklin
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Meaning is seperate from intent.

....

In fact, I think that computers provide a case and point against his over simplified classification. They clearly have meaning even though they do not have consciousness.

On the first point, you’re half right and half wrong. The word “intent” refers both to a purpose, i.e. motivation, and to a mental proposition held by the speaker. Computers clearly have neither. Even though a computer can have something resembling a grammar, it cannot refer; thus it has syntax but no semantics.
However, when I say that computers generate words that have meaning, I am saying that they follow---much as people do---a set of rules by which the words they put up on their screens make reference to some state of affairs in the world.
That isn’t what the words conventionally mean, but rather than fuss about your word choice, I will simply point out that this has nothing to do with meaning. Nobody would say that electromagnetic radiation has meaning or that it generates words, but it does follow rules (aka laws of physics).
To give an example of why this way of thinking about meaning is more precise, consider this popular use of the term meaning: It didn't rain all summer: that means our crops are likely to fail.
I disagree: this way of talking about things is significantly less precise, because it is based on a blurring of the core meaning of meaning. This all comes down to the question “What do you mean by ‘mean’?”. Here are some of the various uses of the word “mean”: “That was no mean accomplishment”; “The mean age of the students is 20.1 years”; “They are so mean to me”; “This will mean the end of his control over us”; “This will mean so much to me”; “I mean to get home before 6:00”; “Keep off the grass! This means you!”; “His losing his job means that he will have to find another”; “Lucky Strike means fine tobacco”; “Those clouds mean rain”; “She doesn’t mean what she said”; “In saying that, she meant that we should leave”; “‘Procrastinate’ means ‘to put things off’”. Clearly, the word doesn’t represent a single concept, it represents multiple concepts. Only the last two senses are really relevant to these philosophical discussions.
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  • 3 months later...

Look, I don't want to get caught up in technical details here, so I will try to start from the beginning and make my points clear. First, you missed an essential condition for my word meaning: Electro-magnetic radiation does not count because it fails to meet my input and representation criteria (the modeled after a particular state of affairs claim.)

However, I will put my case against Peikoff very succintly: Peikoff's logic is inherently constructivist and, therefore, undercuts---without purpose or justification---the epistemological foundations of all math and science. His attempt to avoid Platonism and Nominalism is incorrent (and bordering on the subjective): those tendencies ought to be fought by appealing to natural law.

Despite his claims to the contrary, the law of the excluded middle holds in almost all contexts whether or not there is "evidence". If I say that there are purple men on Mars, this claim is not arbitrary in the same way "angels have pink wings" is. It could be verified empirically and, thus, is either true or false.

If you do not understand where my problems with Peikoff's logic come from, read OPAR and then compare his account of logic with the Kant inspired intuitionistic logic.

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  • 1 year later...

[Edit: It is unnecessarily wasteful to repeat your immediately preceding post. I deleted the repetition D.O.]

I see my comments were not well understood. I think I can put them better now. The question is, Is a computer carrying out a program to perform a given calculation like the bird in Peikoff's writing that says 2+2=4, or is it fundamentally different? My answer is that a consistent causal relationship is as important as consciousness when it comes to meaning and actually has to be present at each step of the process of producing a meaningful statement while consciousness needs only to be present at one point in the process. While the meaning of the symbols the computer gives as output is formed by the humans creating the language of the program, no human is aware of all the specific steps the computer takes in performing its task---only the general character of those steps. The reason that the output of the computer has meaning is that a causal relationship to reality has been maintained throughout. It is this causal relationship that matters. The identifications of conscious minds have meaning because they also necessitate a causal relationship (when the concepts are properly formed). To give another example, if I see a bear's footprint in the snow, I can say that that means a bear has walked there even though no conscious mind produced the print. The meaning is found in the causal relationship between the foot and the snow and only requires the involvment of a consciousness at one point in the chain (namely, when I look at the bear foot print). This may seem like a trivial point, but think about the consequences for computer sciences if the output of unconcsious entities was considered meaningless in all contexts.

If my account seems confused, I welcome questions and comments. My only purpose is to make Peikoff's account of Reason in OPAR consistent with the reality that computers can perform meaningful tasks---i.e. give output that has meaning. (This is what I meant by "produce meaning" before. Give output that has meaning seems clearer though.)

I also want to clarify some of my other posts when I have the time.

[Edit: It is also unnecessarily wasteful to repeat to include the current post in itself. I deleted the repetition. D.O.]

Edited by DavidOdden
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I do not know whether you friend was lying to you, but he certainly is misusing the Axiom of Choice. This axiom was discovered a century ago in the development of set theory, and it continues to engender much discussion amongst mathematicians today. But it usually takes those involved in "philosophy" to distort the issue in a multitude of ways. Simply put, the Axiom of Choice says that given a collection of non-empty sets, you can create a set containing one element from each of the collection by choosing a member from each set. What this means, _within that set theory_, for the real numbers, does not "contradict their nature," but rather, due to a technical issue known as "well-ordering," the standard ordering of the real numbers is not considered to be well-ordered. So that set theory cannot prescribe a definite function to well-order the reals. So what?

Looking at this a year later, I still cannot believe that you did not consider the possibility that I was misquoted. Ashley's account of what I was saying was distorted beyond recognition by the fact that the ideas I was discussing were highly unfamiliar to her. All I said was that the Axiom of choice has the unusual consequence of showing certain things to exist mathematically without necessitating that a concrete example can be given. For example, the proof of the existence of transcendental numbers came before an example of a transcendental number could be given. I simply commented that this had interesting philosophical consequences.

I think you are a disgusting slanderer. You should have considered the possibility that Ashley---though she is very bright---might have been the mistaken one given that she has no serious mathematical experience. I demand an apology---you slander me most unfairly. After a year you never apologized. How could the possibility that she was mistaken never occur to you?

It sounds like he was.

I am furious with you. Ashley mistakenly paraphrased me because the ideas I was discussing were not familiar to her. I can't believe that you did not consider this possibility. I would not use a phrase "contradicts the nature of the real numbers" unless I was possibly trying to show something was clealry false. Again, look at my other posts to see what I was really discussing (i.e. that the axiom of choice leads to proofs of existence without specific examples being available. I am furious and I think your comment is borderline slander.

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A consistent causal relationship is as important as consciousness when it comes to meaning and actually has to be present at each step of the process of producing a meaningful statement while consciousness needs only to be present at one point in the process.
translation for us laymen, please? What are you (a year later) arguing for/against?
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Actually, the quotes were not the same. The later contained the phrase that I was not contradicting Peikoff but simply furthering the identification so that an apparent hostility to computers might be eliminated.

My question was, Is a computer like the Bird that chirps 2+2=4 and therefore meaningless even though a computer is not conscious? I answer no, and I give my reason. Please read the OPAR section on reason and then see my above comment. I admit that this topic is rather far from the axiom of choice question we, but it did eventually develope into this.

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If you do not understand where my problems with Peikoff's logic come from, read OPAR and then compare his account of logic with the Kant inspired intuitionistic logic.

I disagree with the above attempt to slander intuitionistic logic as being "Kantian".

From the standpoint of Topos Theory (which theory is a good contender to replace Set Theory as a foundational theory of mathematics), inituitionistic logic naturally arises as the more general logic on a topos. Classical logic arises on a subset of topoi, and so can only be viewed as a less general logic specific to systems with a certain extra structure.

The rub is that the topoi most used in mathematics and physics until now have come from that special subset wherein classical logic arises.

Anyway, the axioms of Topos Theory arose in the context of Algebraic Geometry, so cannot be rejected as being an arbitrary "Kantian" construct, but rather fell out of structures that were discovered while investigating problems in that field.

To repeat, intuitionistic is a perfectly valid and more general logic than classical logic which ought to be viewed as intuitionistic logic with extra structure (axioms).

Edited by punk
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I do remember what I said which sounded to Ashley like [color="#000000"]contradicts the nature of the reals now. I said that constructivism does not allow for the real numbers to exist, which I found flat wrong given the effectiveness of physical theories using the real numbers. Thus, I said that "Failing to accept the axiom of choice is to contradict nature. The rejection of the real numbers is a sign of this." This is what she did not understand. I believe she overheard a conversation between a brilliant mathematics student named Chris Dodd and me. While Dodd is brilliant, at that time he had not read much on the philosophy of mathematics, so I was explaining the Hilbert (Formalist)/Brouwer (Constructivist) divide to him.

Please delete your earlier comments that insult me; it is only fair.

God, I would hate it if most objectivists familiar with math became followers of those neo-Kantian jackasses like Brouwer---the constructivists.

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  • 6 years later...

This topic has been fairly well picked over and a long time ago. I'd just like to add that the Axiom of Choice asserts the existence of a set and does not specify how to find it. In other words, it offers existence without any means of validation. Such an assertion is contrary to Objectivism. We can easily validate the existence of natural numbers by correspondence with observable properties of physical objects. However, you cannot validate the existence of a choice function for infinite collections of sets. It is pure existence without reduction. This is akin to as assertion of the existence of dragons. However, just because you can define it doesn't mean that it exists. Just as you cannot validate the existence of dragons by reduction to perception, so you cannot validate the existence of the choice function by reduction to anything. In this sense, faith in the Axiom of Choice is a form of mysticism.

 

One is reluctant to give up the Axiom of Choice because it makes certain constructions easy with no clear alternative constructions known. Just because it makes the construction of a theory simple doesn't mean that the theory is philosophically acceptable. Standard math is based on certain mythologies such as the Axiom of Choice. To accept them is to reject reduction--an essential element of objectivism.

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One is reluctant to give up the Axiom of Choice because it makes certain constructions easy with no clear alternative constructions known. Just because it makes the construction of a theory simple doesn't mean that the theory is philosophically acceptable. Standard math is based on certain mythologies such as the Axiom of Choice. To accept them is to reject reduction--an essential element of objectivism.

Correct me if I'm wrong, but doesn't the axiom of choice suggest that any set will always have a function to apply to it in order to select something from that set? I'm not so familiar with math, but I did look into the axiom of choice a few months ago. One definition I found is this one:

 

"Axiom of Choice. Let C be a collection of nonempty sets. Then we can choose a member from each set in that collection. In other words, there exists a function f defined on C with the property that, for each set S in the collection, f(S) is a member of S."

 

http://www.math.vanderbilt.edu/~schectex/ccc/choice.html

 

I don't see why it's problematic for infinite sets, unless you think even infinite set is an invalid concept. We sort of talked about this before. There is no metaphysical infinity, but conceptually there is. There are no metaphysical sets either.  You could certainly validate it to the degree that you are always able to make some choice in any circumstance. If you mean validate as in seeing axioms of choice floating around, of course that is not valid. Can you give an example, in math-speak if you want (I'll try to understand), of when a choice might be impossible - a non-empty set where there *cannot* be a choice? The evidence that it is the axiom of choice itself, but if it is false, then what makes is a possible example that would mean it is false? Certainly, math is needed to demonstrate past intuition that the AoC is indeed true, but accepting it as true does not contradict my knowledge especially since I have a connection to choices, and from there I get a connection to actions.

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When is a choice not a choice? When there is no mechanism conceivable for making that choice. This is exactly what the Axion of Choice asserts. It is akin to my asserting that in the afterlife I will choose between a fairy princess and Xena for mate. Conceptually I may make such a choice but it is meaningless nonsense without a mechanism by which I might realistically make such a choice.

As a mathematical example I offer you unmeasurable sets. Their "construction" requires the AoC, but is anything unmeasurable ever truly real? Keep in mind that we are not talking about just circumstantial incapacity to measure but conceptual incapacity. We are in principle incapable of measuring them. Aren't all perceptions measurments of observable properties of physical objects? In this sense, unmeasurable sets are excluded from any conceivable perception and hence are not subject to reduction. Philosophically, they are dragons. We can define them but not perceive them. (A dragon is a flying reptile that breathes fire.) Any such concept, being in principle removed from reality, in rejected as nonsense.

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As a mathematical example I offer you unmeasurable sets. Their "construction" requires the AoC, but is anything unmeasurable ever truly real?

Depends. What do you mean by (un)measurable? Is it the same sense Rand meant by measurable? To me, unmeasurable set sounds like a type of infinite set, which is valid. Also, to be precise, it is all abstractions that have measurements, and directly perceiving is not necessary on the level for abstractions of abstractions.

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The classical construction of an unmeasurable set constructs a subset of the closed interval [0,1] for which it is shown that it is not possible to assign any reasonable length. The nonmeasurable set, if it had a length, would have length less that or equal to one and greater than or equal to zero. However, no such length is assignable. There are an infinite number of disjoint such sets whose union is the entire interval. However, they cannot have length zero.

Okay, there is an important principle here that you did not address that is really the salient point. If AoC were restricted to finite sets then it would be trivially valid since in that case it is always possible to find a choice procedure. For uncountable collections of uncountable sets some indication of choice procedure is particularly important. What AoC does in this case is disregard the necessity of pointing out howsuch choices may be made and just assumes that such a procedure exists.

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In direct response to your post, second-order concepts must be traceable to first-order concepts that are perceivable. Otherwise, they are invalid. I think AoC applied to infinite sets is an example of a good concept becomming disconnected from an important validating principle by dropping context upon extrapolation to a broader context.

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I took a course in Set Theory about 6 months ago. I admit that I started the course expecting to blow it all out of the water and find contradictions everywhere like you are trying to show aleph_1. However, the course totally changed my opinion of the subject, which in the beginning was pretty low. 

 
The bottom line is this. All mathematical concepts, sets included, are tools for solving problems. They don't exist in reality in the way physical objects do, nor in another dimension as platonists believe. I think that the peculiar thing about mathematical concepts, which has led to so much confusion about what their nature is, is that they are not only very abstract (removed from the perceptual level), but they are also extremely specific. "Circles" don't exist anywhere. What exist are things which take on the shape of a circle. Yet, a "circle" is a very precise thing. It can be defined solely by specifying a point and a radius. Most mathematical concepts are like this. The sets that come up in practice (solving math problems, proving theorems, etc.) are definable in very simple terms. The fact that mathematical concepts are so precise necessitates visual symbolic notation, as opposed to just using words, in order to deal with these concepts while we are solving problems, in order to hold them in our minds. When you do mathematics for long enough, there is a tendency to begin to believe that these symbols you are writing down on your paper are actual "things", like the physical objects we see around us. This is why so many mathematicians become platonists. They strongly feel that sets, numbers, shapes, LIVE somewhere, because they "see" them every day on their blackboards and papers. But the truth of the matter is that these concepts don't live anywhere -- they are just problem solving tools.

 

Now apply this fact to set theory. Mathematicians deal with sets ALOT. They are EXTREMELY useful for proving theorems, solving problems, and understanding mathematics. And before the days of Bertrand Russell, mathematicians believed that, in the course of solving a problem, you could construct any old set you wanted to, just by using set builder notation. However, Bertrand Russell showed that you can construct a "set" using set builder notation which is actually meaningless; that is, there is an element which is both in the set and not in the set at the same time. Obviously, using such a set in proving a theorem would be disastrous, as it would allow you to prove something which is not true....

 

So mathematicians started to realize that a more rigorous definition of a set was needed. They needed to know that the tools they were using would not lead to contradictions in their proofs. When they asked the question, "What is a set?", what they meant was, "What sets can we construct validly, without introducing contradictions into our thought processes?" The answer to this question was axiomatic set theory: a list of basic truths about things that we know are valid sets, and how further valid sets can be constructed. For example, any set containing one element is a valid set, unions of sets are valid sets, the set of all subsets of a set is a valid set, etc. The point of laying down these axioms was to show that all the sets we would ever need in practice can be deduced from these axioms. Since the axioms were clearly based on facts of reality and valid, any set you could construct using the axioms would also have to be valid. However, they came to realize that there were certain sets which they needed to use in practice which could not be shown to be valid from these axioms; there was no way to construct them using the concepts of union, power set, etc. from more trivial sets; for instance, if you had a collection of sets, it made sense that you should be able to construct a set consisting of ordered pairs for which the first element in each ordered pair was a set in the collection, and the second element was some element in that set. In other words, the axiom of choice made sense to them intuitively--it was an obviously valid process of constructing a set out of more trivial sets. But since they could not prove that such a set was valid from the other axioms, they had to assert it, making the axiom of choice an axiom itself. (Unfortunately, since it has been half a year since I took the course, and this is not my specialty, I can't recall any good examples of AC being necessary right now, but the unmeasurable set example that you (aleph_1) gave seems okay.)

 

Anyway, this, in my opinion, is the objective way of looking at set theory. The axiom of choice is not akin to an assertion of dragons. It is merely an observation that a certain type of tool (a set constructed via AC) is valid and does not lead to contradictions in one's thinking. Some famous mathematician (Cantor maybe?) once said that "Existence is freedom from contradiction." While this is obviously wrong if applied loosely to everything in reality...I think there is some truth to this in mathematics. To say that a mathematical concept "exists" in many cases means to say that it is a valid tool, ie. it does not lead to contradictions.

Edited by itsjames
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Near downtown Houston there is a building withthe message boldly painted on its side, "End Hunger". Ok, I choose to end hunger. But wait, where is the mechanism? How? How can I end hunger. A choice without a mechanism for implementation is irrational futility.

Now, I have a finite set and I wish to choose among the elements. How? It is simple. I line the elements up and take the first one in the list. The mechanism exists.

For uncountable sets it is impossible to make such a list. For an infinite collection of uncountable sets, how am I to make a choice? No mechanism is conceivable. AoC asserts that there exists such a choice but provides no mechanism for how to make a choice.

I can program a computer to solve the finite choice problem. However, no computing device could ever solve the arbitrary case.

Just as with hunger, if a mechanism does not exist, then you commit the error of wishful thinking.

Edited by aleph_1
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For uncountable sets it is impossible to make such a list. For an infinite collection of uncountable sets, how am I to make a choice? No mechanism is conceivable. AoC asserts that there exists such a choice but provides no mechanism for how to make a choice.

Why couldn't you merely pick *any* element of the set by arbitrary standards? AoC presumes no empty sets, so it has some member at least, even if not measurable. I could say something like "the first member I happened to notice", or "the last member I happened to notice". My thought is that all you need to do is take some random members of any set, make a subset out of those members, then make any selection as you would with countable sets. Even a noncountable set has distinct members, right, even if the set has no cardinality or any distinctness to the set as a whole? That's my premise.

 

I understand the "end [world] hunger" example, but that in some sense is an empty set. No choice is possible because there is absolutely no member of a set "actions that end [world] hunger". There is literally nothing to select.

 

I find this rather interesting, because it's one of the few math topics that makes math sound really interesting.

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I was wrong. Eiuol is right. The mechanism is provided by AoC itself. Since the Well Ordering Principle is equivalent to AoC, you can also order the set and choose the first element.

I guess my questions really do have to do with sets that are actually infinite and not with the Axion of Choice. Nice discussion.

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I think you are conceding for the wrong reasons. Yes, the well ordering principle is equivalent to AoC, but your line of reasoning would then call for you to ask the question "How do you well order a given set? What is the mechanism?"

 

Your argument seems to be that it is irrational to assert the existence of a "choice" function without giving a concrete example. (Or, it is irrational to assert the existence of a well ordering without giving an example). But the point here is that to assert the existence of a function (which may be considered as a type of set) is very different from asserting the existence of, say, a dragon. Sets and functions are not physical things, they are ways of regarding things. Three toys sitting on the floor in front of you don't constitute a set of toys unless you choose to regard them that way. To say that a set exists is merely to say that it doesn't lead to contradictions once you define it, ie. that everything is either in the set or not in the set, but not both at the same time. 

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I could say something like "the first member I happened to notice"

The language of set theory does not have such terminology as "I happened to notice". It is proven that we cannot prove in ZF the statement "every non-empty set has a choice function", thus to have the result that every non-empty set has a choice function, we have to add it as an axiom to the axioms of ZF. Resorting to terminology that is not even in the language of set theory is not considered a viable solution since the solution mathematicians demand is that of a formulation in the language of set theory.

Edited by GrandMinnow
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I was wrong. Eiuol is right. The mechanism is provided by AoC itself. Since the Well Ordering Principle is equivalent to AoC, you can also order the set and choose the first element.

The well ordering principle is that every non-empty set of natural numbers has a least member. That is not equivalent to the axiom of choice. What you mean is that well ordering theorem: that every non-empty set has a well ordering. But still, your objection to the axiom of choice is that it is non-constructive, yet so is the well ordering theorem. If you object that the axiom of choice provides a choice function but does not show a construction of a choice function, then we might as well observe that the well ordering theorem provides a well ordering but does not show a construction of a well ordering.

Edited by GrandMinnow
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