Announcement: new book being written (by me!)


Roger Bissell

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Stephen Boydstun, on 27 Mar 2013 - 09:41, said:

Hi Roger,

In the significant technical sense for logic and mathematics, sets are defined implicitly in an assembly of axioms, such as these. This approach to set theory yields the desirable result that certain sets that one can say, such as “the set of all sets” or “the set of all cardinal numbers” do not exist. They are not licensed by the axioms. As we see from the ZF axioms, the null set is said to exist in the sense that sets exist. I would suggest considering whether you want to affirm the existence of any sets as implicitly defined by these axioms and as used in logic and mathematics today. Or consider whether you would aim for discerning a different notion of mathematical and logical existence that is some specific sort rarified in comparison to physical-object existence, then see if you want to deny that sort of existence for the null set, but not all others.

Axiomatic set theory is where set theory is in our era. These are sets in a sense more than collection, which latter is a legitimate common usage of the term set, but not what matters, for example, in the set-theoretic axiomatization of group theory or measurement theory. I encourage you to find out what those symbols mean in those axiom statements and be able to read in plain English what those axioms say in their symbolic notation. They are plain English sentences (or plain German, . . .).

Stephen

Thanks, Stephen, for the clarification and the suggestions.

Actually, I ~have~ been working along another avenue. I've been considering number as relational, and I've been considering how the number zero functions in the context of each of the types of numbers (counting #'s, integers, real #'s). If I interpret your use of the term "rarified" correctly, I've been doing what you suggest -- seeing numbers as abstractions of relationships between real things.

I don't want to throw out the accurate but mislabeled/misconceptualized results of modern math, any more than I want to throw out the accurate but mislabeled/misconceptualized results of modern physics. I just want to understand them in the same manner that I understand abstractions more generally, as per Rand's theory of concepts -- and place them in my knowledge hierarchy accordingly.

I've been listening to (ARI lecturer) Pat Corvini's lectures on number (total of 6, over 2 years), and she has a lot of insights, though she does not discuss zero specifically. She does, however, address the issue of the supposed one-to-one correspondendence between infinite "sets," and she pretty well demolishes the modern notion that there is, for instance, the same number of counting numbers as even numbers. (We've discussed that here previously, to no avail.)

Thanks again.

REB

Zero is the identity element of the additive semi-group of integers.

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Roger, I think you are on a promising line of thought to develop. I very much recommend along similar lines John Bigelow’s The Reality of Numbers (1988). You can get it used for about $50 at ABE Books or look for it in a library nearest you using World Cat. Bigelow characterizes the natural numbers as relations and all the numbers beyond those (integer, rational, real, imaginary, complex) as various relations between relations. His work can speed you up I bet, and anyway whatever wheel you make, let it be set out explicitly how it is related to Bigelow’s wheel. –S

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Stephen Boydstun, on 27 Mar 2013 - 14:56, said:

.

Roger, I think you are on a promising line of thought to develop. I very much recommend along similar lines John Bigelow’s The Reality of Numbers (1988). You can get it used for about $50 at ABE Books or look for it in a library nearest you using World Cat. Bigelow characterizes the natural numbers as relations and all the numbers beyond those (integer, rational, real, imaginary, complex) as various relations between relations. His work can speed you up I bet, and anyway whatever wheel you make, let it be set out explicitly how it is related to Bigelow’s wheel. –S

Thanks very much for the recommendation, Stephen. Your erudition is often helpful and always amazing. :-)

REB

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I've been listening to (ARI lecturer) Pat Corvini's lectures on number (total of 6, over 2 years), and she has a lot of insights, though she does not discuss zero specifically. She does, however, address the issue of the supposed one-to-one correspondendence between infinite "sets," and she pretty well demolishes the modern notion that there is, for instance, the same number of counting numbers as even numbers. (We've discussed that here previously, to no avail.)

1. What is her argument that the number of counting numbers and even numbers are not the same? (I do agree.)

2. Do you have a URL where was it discussed?

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I've been listening to (ARI lecturer) Pat Corvini's lectures on number (total of 6, over 2 years), and she has a lot of insights, though she does not discuss zero specifically. She does, however, address the issue of the supposed one-to-one correspondendence between infinite "sets," and she pretty well demolishes the modern notion that there is, for instance, the same number of counting numbers as even numbers. (We've discussed that here previously, to no avail.)

Thanks again.

REB

Corvin's statement is dead wrong. The one to one correspondence between the set of integers and the set of even integers is established by the function n <-> 2*n. It is 1-1 and onto. End of Proof.

If I were charged money to hear a lecture like that I would demand a refund on the ground that the instructor was mathematically incompetent.

Ba'al Chatzaf

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Merlin Jetton, on 27 Mar 2013 - 20:50, said:

Roger Bissell, on 27 Mar 2013 - 10:57, said:

I've been listening to (ARI lecturer) Pat Corvini's lectures on number (total of 6, over 2 years), and she has a lot of insights, though she does not discuss zero specifically. She does, however, address the issue of the supposed one-to-one correspondendence between infinite "sets," and she pretty well demolishes the modern notion that there is, for instance, the same number of counting numbers as even numbers. (We've discussed that here previously, to no avail.)

1. What is her argument that the number of counting numbers and even numbers are not the same? (I do agree.)

2. Do you have a URL where was it discussed?

2. No. These are taped lectures that are sold by the Ayn Rand Bookstore (or whatever ARI calls it these days). The set that contains this discussion is "2, 3, 4, and all that--the sequel" (2008). Corvini discusses the issue of one-to-one correspondence extensively in lecture 1 and gives her solution in lecture 3.

1. I'm not going to present Corvini's argument here. She is working on a book, and I invite you and anyone else interested to purchase the lectures...or to wait for the book. However, I would note that anyone who thinks that the method of pairing can be applied to a situation with no definite numerosity and in which the pair operation cannot come to an end...well, that person needs to check his premises. Also, I will offer my own observation: between 1 and 10, there are twice as many counting numbers as even numbers, between 11 and 20...the same, and so on. There is no evidence and no reason to suspect that this ratio ever changes. it is not a converging series.

For Ba'al: any purported argument that the ratio of counting numbers to even numbers is 1:1 needs to provide something it cannot: evidence or logic. (A claim based on the results of a pairing operation which in practice cannot demonstrate any scintilla of an iota of an indication that the repeatedly observed 2:1 ratio even twitches minutely in the direction of the purported 1:1 ratio, and which by definition cannot even come to an end, does not qualify as evidence or logic.)

REB

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For Ba'al: any purported argument that the ratio of counting numbers to even numbers is 1:1 needs to provide something it cannot: evidence or logic. (A claim based on the results of a pairing operation which in practice cannot demonstrate any scintilla of an iota of an indication that the repeatedly observed 2:1 ratio even twitches minutely in the direction of the purported 1:1 ratio, and which by definition cannot even come to an end, does not qualify as evidence or logic.)

REB

The notion of a -ratio- does not enter into the argument of whether a 1-1 correspondences exists or not. You have not clearly understood the definition of equivalent cardinality of sets. I will try one more time to make it clear. Two sets have the same cardinality if and only if there exists a one to one function that maps the first set onto the second. One to One means distinct element have distinct images under the function. Onto means that for each element of the second set there exist an element of the first set that gets mapped into it by the function.

You will note that the word -ratio- does NOT occur in the definition nor is it implied by the definition.

If am sure Corvini's book will be as useful a contribution to the field as was Hatman's book.

Ba'al Chatzaf

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Roger, you wrote, "We've discussed that here previously, to no avail." I interpreted that to mean discussed on OL. Your claim was inaccurate?

y = x if x even, x+1 if x odd. That is a 2-to-1 map, implying X (all integers) has twice as many numbers as Y (even integers).

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Roger, you wrote, "We've discussed that here previously, to no avail." I interpreted that to mean discussed on OL. Your claim was inaccurate?

y = x if x even, x+1 if x odd. That is a 2-to-1 map, implying X (all integers) has twice as many numbers as Y (even integers).

wrong. The integers can be put into 1 - 1 correspondence with the even integers. That means the set of integers and the set of even integers have the same cardinality.

I used to wonder why there are virtually no Objectivists in the front ranks of physics and mathematics. Now I no longer wonder.

Ba'al Chatzaf

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wrong. The integers can be put into 1 - 1 correspondence with the even integers. That means the set of integers and the set of even integers have the same cardinality.

Unstated false premise -- the only way to compare quantities is 1-1 correspondence.

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wrong. The integers can be put into 1 - 1 correspondence with the even integers. That means the set of integers and the set of even integers have the same cardinality.

Unstated false premise -- the only way to compare quantities is 1-1 correspondence.

The definition of equal cardinality is this: A and B have the same cardinality if and only there is a 1 - 1- onto map from A to B.

This works with both finite sets and infinite sets.

There are other ways of comparing sets. For example A is a subset of B if and only if is x is an element of A implies x is an element of B

In this case, the even integers are a proper subset of the the integers.

You might want to learn some mathematics when you get the chance.

ba'al Chatzaf

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In this case, the even integers are a proper subset of the the integers.

You might want to learn some mathematics when you get the chance.

ba'al Chatzaf

Check your premises. You are clinging to a false one.

The following applies with A = even integers and B = all integers.

"If A is a subset of B, but A is not equal to B (i.e. there exists at least one element of B not contained in A), then

  • A is also a proper (or strict) subset of B; this is written as 50de602089f264acf40e7c74aa0a8c49.png " (link)
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In this case, the even integers are a proper subset of the the integers.

You might want to learn some mathematics when you get the chance.

ba'al Chatzaf

Check your premises. You are clinging to a false one.

The following applies with A = even integers and B = all integers.

"If A is a subset of B, but A is not equal to B (i.e. there exists at least one element of B not contained in A), then

  • A is also a proper (or strict) subset of B; this is written as 50de602089f264acf40e7c74aa0a8c49.png " (link)

I pointed out there are other comparisons of sets beside cardinal equivalence. The subset relation is just such an other comparison. Given A and B sets of the same cardinality, A may well be a proper subset of B . In fact that is the definition of an infinite set. A set S is infinite if it has a proper subset T such that T has the same cardinality as S. Which is to say there is a 1 - 1 onto mapping from T to S. The fact that the set of integers has the same cardinality as the set of even integers (a proper subset) shows that the set of integers is infinite. That is an example often used in Set Theory 101.

ba'a Chatzaf

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Merlin Jetton, on 28 Mar 2013 - 06:23, said:

Roger, you wrote, "We've discussed that here previously, to no avail." I interpreted that to mean discussed on OL. Your claim was inaccurate?

y = x if x even, x+1 if x odd. That is a 2-to-1 map, implying X (all integers) has twice as many numbers as Y (even integers).

Merlin, we did at least briefly touch on this issue (# of even #'s =/not= # of counting #'s) sometime in the past 2-3 years. Judging from your position on this, you were not tuned in at the moment, or you would have jumped in when Ba'al was doing his usual Cantorian schtick.

It's obvious to me that (1) inductively, since for all finite intervals, the number of counting numbers is twice that of the number of even numbers, there is no reason -- other than Cantor's arbitrary misapplication of one-one pairing -- to believe that the number of ~all~ counting numbers should ~not~ be twice that of the number of all even numbers; (2) the technique/operation of pairing numbers is derived from concrete reality for the purpose of comparing groups that have a definite numerosity, and therefore is understood as a process that ~has a definite end~.

Not only is counting the number of items in an infinite set impossible, but so is comparing the number of items in ~two~ such sets. You can match sheep against pebbles and eventually come up with a determination of whether they have the same or a different number, because the process of matching is finite. You cannot do this with the counting and even numbers. It is not really a one-to-one matching, because you don't/can't finish it, and each time you check your progress, you see that the matching has not budged from a two-to-one matching and gives no rational indication that it will.

In other words, Cantor (and his followers, some of whom may be reading this post!) have dropped the cognitive context of the matching operation. The notion of a "one-to-one correspondence" between the counting and even numbers is, in Randian terms, a "stolen concept."

There are various reasons why no Objectivist has yet been in the forefront of math and science, but misunderstanding Cantor is ~not~ one of those reasons! One of the actual reasons is that modern math and science are such an incredible conceptual mess, that it would take decades for one really focused good thinker to undo the errors -- to understand the false premises of the mainstreamers' arguments, and to formulate the correct replacement conceptual framework.

If you have noticed, most Randian philosophical work has been done in ethical and political theory. This is not an accident. Although ethics and politics are logically/hierarchically dependent upon metaphysics and epistemology (and science, among other things), they are also closer to people's experience and concerns, and are experienced by most as being THE crucial things to understand and get right.

But as radical and clear as the work in these areas by Objectivists (and libertarians) has been, egoists and capitalists are ~still~ subject to the slings and errors and smears of those who fear and hate those who threaten their altruist/collectivist rackets. In fact, if anything, the condemnation and ridicule have intensified, even as the body of work and the number of pro-individualist thinkers and think tanks has proliferated. This kind of belittlement of those one disagrees with is not just a phenomenon of the ethical-political realm either, as we have seen here on OL. [written after casting a baleful glance at earlier posts to this thread]

REB

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Here's another way of seeing (through) the flawed thinking of Cantor's one-to-one of counting and even numbers.

1. Suppose I decide to start by matching the first 10 counting numbers to the first 10 even numbers. I note that I have used counting numbers up to 10, but even numbers up to 20. However, this is misleading. Even though in ~matching~ the first 10 evens with counting numbers, I only had to use the counting numbers up to 10, in ~identifying~ the first 10 even numbers, I have had to use the counting numbers up to 20. The first 10 even numbers are ~only derivable from~ the first 20 counting numbers. In other words, I have had to ~make use of~ the first 20 counting numbers ~in order to generate~ the first 10 even numbers. So, simply staring at the one-to-one correspondence is very deceptive. The second 10 counting numbers are already in the game, even though they're not explicitly laid out in the pairing process.

2. Inductively, this is true for ANY finite or infinite set of counting and even numbers. Even though in ~matching~ the first N evens with counting numbers, I only have to use the counting numbers up to N, in ~identifying~ the first N even numbers, I have to use the counting numbers up to 2N. The first N even numbers are ~only derivable from~ the first 2N even numbers. In other words, I have to ~make use of~ the first 2N counting numbers ~in order to generate~ the first N even numbers. So, again, simply staring at the one-to-one correspondence, as this time it trails off into the infinite distance, is very deceptive. The second N counting numbers are already in the game, even though they're not explicitly laid out in the infinite, one-to-one, pairing process.

The flaw in Cantor's gimmick is intuitively obvious to an intelligent high school student. What is intuited is that there is some smuggling going in, some unacknowledged use of numbers that are then disowned in analyzing and evaluating the pairing. If you use the second N counting numbers, and then deny that you are using them, you are committing the fallacy of equivocation.

Or, again in Randian terms, you are committing the stolen concept fallacy: you are using one aspect of the nature of the second N counting numbers in order to deny another aspect of their nature. The second N counting numbers are needed in order to generate the first N even numbers, and then they are discarded from the pairing, as though their previously necessary existence did not establish that there were twice as many counting numbers as even numbers. In even more simple Randian terms, you are biting the mathematical hand that feeds you. :-)

REB

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A ~proper~ one-to-one correspondence between the counting and even numbers would note that for every even number, there correspond exactly ~two~ counting numbers, the two needed to ~generate~ that even number -- namely, 2n - 1 and 2n.

Specifically, it would make explicit the fact that in generating the even number 2, you are using the counting numbers 1 and 2 -- that in generating the even number 4, you are using the counting numbers 3 and 4 -- that in generating the even number 6, you are using the counting numbers 5 and 6 -- etc. And that what you are NOT doing is using the counting number 2 to generate the even number 4, the counting number 3 to generate the even number 6, etc.

This correspondence keeps the game honest. It acknowledges that ~two~ counting numbers are needed to generate every even number. It does not pretend that one of them does not exist, in order to make the bogus argument that there are as many even numbers as counting numbers.

It avoids the larcenous gimmick -- or incompetent error -- that Cantor introduced over a century ago.

I can't help but note the similarity between this issue and the fact that while every child has two parents, it has become fashionable for people to try (in Cantoresque fashion) to pretend that the father does not exist. I wonder if Cantor was a radical feminist. :-)

REB

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A ~proper~ one-to-one correspondence between the counting and even numbers would note that for every even number, there correspond exactly ~two~ counting numbers, the two needed to ~generate~ that even number -- namely, 2n - 1 and 2n.

Specifically, it would make explicit the fact that in generating the even number 2, you are using the counting numbers 1 and 2 -- that in generating the even number 4, you are using the counting numbers 3 and 4 -- that in generating the even number 6, you are using the counting numbers 5 and 6 -- etc. And that what you are NOT doing is using the counting number 2 to generate the even number 4, the counting number 3 to generate the even number 6, etc.

This correspondence keeps the game honest. It acknowledges that ~two~ counting numbers are needed to generate every even number. It does not pretend that one of them does not exist, in order to make the bogus argument that there are as many even numbers as counting numbers.

It avoids the larcenous gimmick -- or incompetent error -- that Cantor introduced over a century ago.

I can't help but note the similarity between this issue and the fact that while every child has two parents, it has become fashionable for people to try (in Cantoresque fashion) to pretend that the father does not exist. I wonder if Cantor was a radical feminist. :-)

REB

Whatever happened to multiplying an integer by 2. Did I miss getting the e-mail?

Before pontificating on what is proper and improper about integers, I suggest you learn the Peano axioms for arithmetic.

Please see: http://en.wikipedia.org/wiki/Peano_Axioms

Ba'al Chatzaf

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BaalChatzaf, on 28 Mar 2013 - 20:51, said:

Roger Bissell, on 28 Mar 2013 - 19:04, said:

A ~proper~ one-to-one correspondence between the counting and even numbers would note that for every even number, there correspond exactly ~two~ counting numbers, the two needed to ~generate~ that even number -- namely, 2n - 1 and 2n.

Specifically, it would make explicit the fact that in generating the even number 2, you are using the counting numbers 1 and 2 -- that in generating the even number 4, you are using the counting numbers 3 and 4 -- that in generating the even number 6, you are using the counting numbers 5 and 6 -- etc. And that what you are NOT doing is using the counting number 2 to generate the even number 4, the counting number 3 to generate the even number 6, etc.

This correspondence keeps the game honest. It acknowledges that ~two~ counting numbers are needed to generate every even number. It does not pretend that one of them does not exist, in order to make the bogus argument that there are as many even numbers as counting numbers.

It avoids the larcenous gimmick -- or incompetent error -- that Cantor introduced over a century ago.

I can't help but note the similarity between this issue and the fact that while every child has two parents, it has become fashionable for people to try (in Cantoresque fashion) to pretend that the father does not exist. I wonder if Cantor was a radical feminist. :-)

REB

Whatever happened to multiplying an integer by 2. Did I miss getting the e-mail?

Before pontificating on what is proper and improper about integers, I suggest you learn the Peano axioms for arithmetic.

Please see: http://en.wikipedia.org/wiki/Peano_Axioms

Ba'al Chatzaf

You want multiplication by 2? Fine. But I will remind you that multiplying is based on adding, which is based on counting, so you shouldn't be too surprised to find that my analysis and reply reduce to an illustration based on the sequence of counting numbers..

1 x 2 ~IS~ the last number of the sequence 1 2 [stop]

2 x 2 ~IS~ the last number of the sequence 1 2 [stop] + the last number of the sequeence 1 2 [stop], which in turn ~IS~ the last number of the sequence 1 2 3 4 [stop].

3 x 2 ~IS~ the last number of the sequence 1 2 3 [stop] + the last number of the sequence 1 2 3 [stop], which in turn ~IS~ the last number of the sequence 1 2 3 4 5 6 [stop].

That is why the one-to-one correspondence has to look like ~THIS~:

Counting numbers: 1 2...3 4...5 6 etc.

Even numbers: 2...4...6 etc.

~NOT LIKE THIS CANTORIAN CON-JOB~:

Counting numbers: 1...2...3...4...5...6 etc.

Even numbers: 2...4...6...8...10...12 etc.

Now, in light of this, if you could bring yourself to take a five-minute break from your obsessive insults and condescension and waving of celebrity mathematicians' names at me, you might benefit from re-reading my previous posts.

If you really don't get this, and if you really think that my arguments are null and void if I don't frame them in terms of Peano's axioms, then we're not going to get anywhere.

I'm not optimistic. Please prove me wrong.

REB

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That is why the one-to-one correspondence has to look like ~THIS~:

Counting numbers: 1 2...3 4...5 6 etc.

Even numbers: 2...4...6 etc.

That is a 2-to-1 correspondence (map) in my book, like I described in #58.

Your correspondence is not the one that establishes the equality of the cardinal number of integers and the cardinal number of even integers. There are many mappings from one set to the other. Some of them are NOT 1 - 1 and some of the are not onto.

All that equality of cardinality requires is at least one mapping that is 1 - 1 and maps the first set onto the second.

Ba'al Chatzaf

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Merlin Jetton, on 29 Mar 2013 - 06:25, said:

Roger Bissell, on 28 Mar 2013 - 23:10, said:

That is why the one-to-one correspondence has to look like ~THIS~:

Counting numbers: 1 2...3 4...5 6 etc.

Even numbers: 2...4...6 etc.

That is a 2-to-1 correspondence (map) in my book, like I described in #58.

You are correct, of course, Merlin. It is indeed a two-to-one correspondence. That's why there are twice as many counting numbers as even numbers

The one-to-one correspondence of Cantor proves nothing. As I explained previously, if you use multiplication by 2 to generate the evens, it is just a shell game, a rather lame con job on unsuspecting students (and all too many academics).

Rather than multiplying by two, the simpler way of generating the evens is just to sing out every other number in the counting number sequence: (one) TWO (three) FOUR (five) SIX...etc. Then you can see clearly that instead of "getting" four from two, you get it (in sequence) from three, just as you "get" two in sequence from one. Similarly, instead of "getting" six from three, you get it (in sequence) from five -- and eight (in sequence) from seven. This is the simplest way I have come up with so far of demonstrating how you can see by inspection that you will always have twice as many counting numbers as evens.

REB

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Another way to reason about this is to imagine you have all the integers in a container. Then you remove half of them -- the odd ones. It is absurd to claim there are as many integers after as before. Yet that is in effect what Ba'al and Cantor do. You could equivalently imagine putting a divider in the container, putting the odds on one side and the evens on the other. It is absurd to claim there are as many integers on one side of the divider as on both sides. Either absurd claim violates part-whole logic. That one can compare quantities only by one 1-to-1 correspondence is a false premise.

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Another way to reason about this is to imagine you have all the integers in a container. Then you remove half of them -- the odd ones. It is absurd to claim there are as many integers after as before. Yet that is in effect what Ba'al and Cantor do. You could equivalently imagine putting a divider in the container, putting the odds on one side and the evens on the other. It is absurd to claim there are as many integers on one side of the divider as on both sides. Either absurd claim violates part-whole logic. That one can compare quantities only by one 1-to-1 correspondence is a false premise.

You use the phrase "as many" I use the phrase "have the same cardinality". The phrases are not equivalent. Two non empty sets A, B have the same cardinality if and only if there exists a function f: A -> B such that f is 1 - 1 and f maps A onto B. That is the definition of "same cardinality" that has been used by mathematicians for over 100 years.

At no time did I assert that the set of even integers were not a proper subset of the set of integers.

You phraseology (which is incorrect) applies only to finite sets of objects. The set of integers is an infinite set.

I will now give you the standard definition of an infinite set. a set S is infinite if and only if it has a proper non empty subset T such that S and T have the same cardinality.

I wish you would study set theory carefully before you blurt out your irrelevant Objectivist objections.

At one time I wondered by there are few if any Objectivist in the front lines of theoretical physics or mathematics.

I no longer wonder.

Ba'al Chatzaf

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Merlin Jetton, on 30 Mar 2013 - 07:51, said:

Another way to reason about this is to imagine you have all the integers in a container. Then you remove half of them -- the odd ones. It is absurd to claim there are as many integers after as before. Yet that is in effect what Ba'al and Cantor do. You could equivalently imagine putting a divider in the container, putting the odds on one side and the evens on the other. It is absurd to claim there are as many integers on one side of the divider as on both sides. Either absurd claim violates part-whole logic. That one can compare quantities only by one 1-to-1 correspondence is a false premise.

Good analogy, Merlin. Ba'al and Cantor are, in effect, claiming that when the infinite glass is half full, it's full. :-)

(But also, that when it's full, it's also half full!)

REB

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Merlin Jetton, on 30 Mar 2013 - 07:51, said:

Another way to reason about this is to imagine you have all the integers in a container. Then you remove half of them -- the odd ones. It is absurd to claim there are as many integers after as before. Yet that is in effect what Ba'al and Cantor do. You could equivalently imagine putting a divider in the container, putting the odds on one side and the evens on the other. It is absurd to claim there are as many integers on one side of the divider as on both sides. Either absurd claim violates part-whole logic. That one can compare quantities only by one 1-to-1 correspondence is a false premise.

Mysterious indeed are the ways of The Infinite. The Infinite Part and the Infinite Whole are a Holy, Inscrutable Mystery, not to be questioned or understood by mere mortals, who think in terms of finite parts and wholes. Two in One, One in Two, the Holy Infinite Duality...they are and are not the same.

Cantor was a mystic, who believed his ideas on transfinite numbers to have been communicated to him by God. (See Dauben 2004.) This is all starting to make sense now.

REB

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