How can induction be valid?

[George H. Smith]

You claim that false premises in a deductive argument ~usually~ yield a false conclusion, even if the inference itself is formally valid.

Is this really true? I wonder whether instead it is perhaps 50-50 whether that is the case....

It's funny you should mention this. I haven't studied syllogisms in any detail for many years -- the A, E, I, O stuff -- so I wasn't at all sure how frequently false premises will yield true conclusions. (The example that came to my mind was: All nuns are red; this apple is a nun; therefore this apple is red.)

I was going to drag out yet another book on logic and look into this issue,but I decided to take a short cut by saying that false premises will usually yield false conclusions.

I hoped to slip this by. Then I thought, "Someone will probably call me on this." Sure enough, someone did.

Ghs

[kiaer.ts]

You claim that false premises in a deductive argument ~usually~ yield a false conclusion, even if the inference itself is formally valid.

Is this really true? I wonder whether instead it is perhaps 50-50 whether that is the case....

It's funny you should mention this. I haven't studied syllogisms in any detail for many years -- the A, E, I, O stuff -- so I wasn't at all sure how frequently false premises will yield true conclusions. (The example that came to my mind was: All nuns are red; this apple is a nun; therefore this apple is red.)

I was going to drag out yet another book on logic and look into this issue,but I decided to take a short cut by saying that false premises will usually yield false conclusions.

I hoped to slip this by. Then I thought, "Someone will probably call me on this." Sure enough, someone did.

Ghs

Most of the conclusions will be false.

The Red Nun example works because it is implictly designed backwards from the putatively true conclusion that apples are red. But if we work backward from a semi-randomly chosen conclusion, say that apples are black or blue or white, we will get arguments like All nuns are blue, this apple is a nun, therefore this apple is blue. The ratio of non-red to red colors approaches infinity. Truly randomly generated syllogisms with false premises will almost always generate false conclusions in a ratio of false to true approaching infinity.

Edited September 12, 2010 by Ted Keer

[BaalChatzaf]

That's the real lesson of deduction. Unless you know your premises to be true and your inference to be valid, your conclusions really do not have epistemic status at all. They are just verbal junk.

REB

yup. The only deductions one can take to the bank are sound arguments. Arguments starting with true premises and with the conclusion correctly (validly) inferred from the premises.

Ba'al Chatzaf

[Dennis Hardin]

The error in the Black Swans example consists in the conclusion that color is an essential attribute somehow connected to the nature of Swans. That is an inductive hypothesis based on enumeration, not an inductive inference based on an analysis of the nature of Swans.

Didn't I just say that?

http://www.youtube.com/watch?v=29_VEGPTxGk

[Roger Bissell]

You claim that false premises in a deductive argument ~usually~ yield a false conclusion, even if the inference itself is formally valid.

Is this really true? I wonder whether instead it is perhaps 50-50 whether that is the case....

It's funny you should mention this. I haven't studied syllogisms in any detail for many years -- the A, E, I, O stuff -- so I wasn't at all sure how frequently false premises will yield true conclusions. (The example that came to my mind was: All nuns are red; this apple is a nun; therefore this apple is red.)

I was going to drag out yet another book on logic and look into this issue,but I decided to take a short cut by saying that false premises will usually yield false conclusions.

I hoped to slip this by. Then I thought, "Someone will probably call me on this." Sure enough, someone did.

Ghs

Most of the conclusions will be false.

The Red Nun example works because it is implictly designed backwards from the putatively true conclusion that apples are red. But if we work backward from a semi-randomly chosen conclusion, say that apples are black or blue or white, we will get arguments like All nuns are blue, this apple is a nun, therefore this apple is blue. The ratio of non-red to red colors approaches infinity. Truly randomly generated syllogisms with false premises will almost always generate false conclusions in a ratio of false to true approaching infinity.

Well, Ted, you have reopened the infinity topic I saw, but didn't comment on, from another thread.

I'm sure you're right, that you can generate an infinite number of "false" (this apple is blue) conclusions in a syllogism with one or two false premises. But why isn't it equally clear that you can also generate an infinite number of "true" (this apple is red) conclusions in a syllogism with one or two false premises? Just plug in any one of an infinite number of concepts in place of "nun" (e.g., pope, tree, ocean, microbe, galaxy, etc. etc.), and you get: All X's are red, this apple is an X, therefore this apple is red.

We could argue over which infinite set of "false" conclusion syllogisms was larger -- and that would take us back to the issue of whether there is such a thing as orders of infinity, etc. But really, do we need to argue that?

Perhaps so. Some argue, with the moderns, that the number of odd integers is the same as the number of integers, because you can establish a one-to-one correspondence between them. IM(non-H)O opinion, this is balderdash. If the number of odd integers between 1 and 10 is five, and the number of integers between 1 and 10 is ten, and that ratio of 1:2 remains constant for every observable or conceivable span of ten integers (and why wouldn't it?), then how can that ratio suddenly (so to speak) become 1:1 when the span of integers becomes infinitely large? There is no rational or empirical reason to think it would. Thus, there is a huge intellectual corruption near the base of much of modern mathematics.

But I digress! Mainly, I just wanted to say that if Ted could show/prove that there is some greater "order of infinity" operating when "false" conclusions are produced in valid inference, I'd love to see that demonstration!

REB

[kiaer.ts]

You claim that false premises in a deductive argument ~usually~ yield a false conclusion, even if the inference itself is formally valid.

Is this really true? I wonder whether instead it is perhaps 50-50 whether that is the case....

It's funny you should mention this. I haven't studied syllogisms in any detail for many years -- the A, E, I, O stuff -- so I wasn't at all sure how frequently false premises will yield true conclusions. (The example that came to my mind was: All nuns are red; this apple is a nun; therefore this apple is red.)

I was going to drag out yet another book on logic and look into this issue,but I decided to take a short cut by saying that false premises will usually yield false conclusions.

I hoped to slip this by. Then I thought, "Someone will probably call me on this." Sure enough, someone did.

Ghs

Most of the conclusions will be false.

The Red Nun example works because it is implictly designed backwards from the putatively true conclusion that apples are red. But if we work backward from a semi-randomly chosen conclusion, say that apples are black or blue or white, we will get arguments like All nuns are blue, this apple is a nun, therefore this apple is blue. The ratio of non-red to red colors approaches infinity. Truly randomly generated syllogisms with false premises will almost always generate false conclusions in a ratio of false to true approaching infinity.

Well, Ted, you have reopened the infinity topic I saw, but didn't comment on, from another thread.

I'm sure you're right, that you can generate an infinite number of "false" (this apple is blue) conclusions in a syllogism with one or two false premises. But why isn't it equally clear that you can also generate an infinite number of "true" (this apple is red) conclusions in a syllogism with one or two false premises? Just plug in any one of an infinite number of concepts in place of "nun" (e.g., pope, tree, ocean, microbe, galaxy, etc. etc.), and you get: All X's are red, this apple is an X, therefore this apple is red.

We could argue over which infinite set of "false" conclusion syllogisms was larger -- and that would take us back to the issue of whether there is such a thing as orders of infinity, etc. But really, do we need to argue that?

Perhaps so. Some argue, with the moderns, that the number of odd integers is the same as the number of integers, because you can establish a one-to-one correspondence between them. IM(non-H)O opinion, this is balderdash. If the number of odd integers between 1 and 10 is five, and the number of integers between 1 and 10 is ten, and that ratio of 1:2 remains constant for every observable or conceivable span of ten integers (and why wouldn't it?), then how can that ratio suddenly (so to speak) become 1:1 when the span of integers becomes infinitely large? There is no rational or empirical reason to think it would. Thus, there is a huge intellectual corruption near the base of much of modern mathematics.

But I digress! Mainly, I just wanted to say that if Ted could show/prove that there is some greater "order of infinity" operating when "false" conclusions are produced in valid inference, I'd love to see that demonstration!

REB

Good to see your presence here, Roger.

I do accept that some infinities are "larger" than others based on the ability or not to establish a one-to-one correspondence. The matter is simply stipulated, which is fine so far as it goes. I see how you could balk at the "equation" of the odds with the integers, but it follows from the definitions being used. I would say that equality is simply the wrong word (we need a new one, since the sense differs) and that you cannot assert that the amounts are ''actually'' equal, since there are no actual infinities.

I would go about the ratio of true to false propositions by looking at the fact that you have to design the syllogism backwards from the desired conclusion to have any real chance of getting a true conclusion from false premises. Don't tell me you came up with yours without careful planning. Just try sticking a list of ten fruits and ten colors in a barrel and seeing if fifty percent of the arguments you create by this ad lib process lead to true conclusions.

The class of the non-red is always infinitely bigger than the class of the red.

[BaalChatzaf]

Perhaps so. Some argue, with the moderns, that the number of odd integers is the same as the number of integers, because you can establish a one-to-one correspondence between them. IM(non-H)O opinion, this is balderdash. If the number of odd integers between 1 and 10 is five, and the number of integers between 1 and 10 is ten, and that ratio of 1:2 remains constant for every observable or conceivable span of ten integers (and why wouldn't it?), then how can that ratio suddenly (so to speak) become 1:1 when the span of integers becomes infinitely large? There is no rational or empirical reason to think it would. Thus, there is a huge intellectual corruption near the base of much of modern mathematics.

The set of integers can be put into 1-1 correspondence with the set of even integers:

Behold! n <-> 2*n

For each integer an even integer(just multiply by 2). For each even integer an integer (just divide by 2). Why do you find this problematical?

However it is meaningless to talk about the ratio of cardinality of infinite sets. Aleph-0 over Aleph-0 is not a fraction and not a ratio.

Ba'al Chatzaf

[Roger Bissell]

Perhaps so. Some argue, with the moderns, that the number of odd integers is the same as the number of integers, because you can establish a one-to-one correspondence between them. IM(non-H)O opinion, this is balderdash. If the number of odd integers between 1 and 10 is five, and the number of integers between 1 and 10 is ten, and that ratio of 1:2 remains constant for every observable or conceivable span of ten integers (and why wouldn't it?), then how can that ratio suddenly (so to speak) become 1:1 when the span of integers becomes infinitely large? There is no rational or empirical reason to think it would. Thus, there is a huge intellectual corruption near the base of much of modern mathematics.

The set of integers can be put into 1-1 correspondence with the set of even integers:

Behold! n <-> 2*n

For each integer an even integer(just multiply by 2). For each even integer an integer (just divide by 2). Why do you find this problematical?

However it is meaningless to talk about the ratio of cardinality of infinite sets. Aleph-0 over Aleph-0 is not a fraction and not a ratio.

Ba'al Chatzaf

Ba'al -- if there is a one-to-one correspondence between the members of two sets, doesn't that mean that the two sets have the same number of members? If not, then what is the sense and worth of a one-to-one correspondence in number theory and algebra? But if so, then my question stands: how can there be as many odd integers as there are integers? Forget the talk of fractions and ratios. If each span of ten integers has twice as many integers as odd integers, at what point does the number of odd integers in an interval begin to exceed the number of integers in that interval, so that the total of odd integers can be equal to (in one-to-one correspondence) with the integers?

Sorry. There ~has~ to be twice as many integers as odd integers. If it is so for any finite range of integers (granting it has an even number of integers), then it has to be so for an infinite range of integers. One-to-one correspondence notwithstanding.

REB

[kiaer.ts]

Perhaps so. Some argue, with the moderns, that the number of odd integers is the same as the number of integers, because you can establish a one-to-one correspondence between them. IM(non-H)O opinion, this is balderdash. If the number of odd integers between 1 and 10 is five, and the number of integers between 1 and 10 is ten, and that ratio of 1:2 remains constant for every observable or conceivable span of ten integers (and why wouldn't it?), then how can that ratio suddenly (so to speak) become 1:1 when the span of integers becomes infinitely large? There is no rational or empirical reason to think it would. Thus, there is a huge intellectual corruption near the base of much of modern mathematics.

The set of integers can be put into 1-1 correspondence with the set of even integers:

Behold! n <-> 2*n

For each integer an even integer(just multiply by 2). For each even integer an integer (just divide by 2). Why do you find this problematical?

However it is meaningless to talk about the ratio of cardinality of infinite sets. Aleph-0 over Aleph-0 is not a fraction and not a ratio.

Ba'al Chatzaf

Ba'al -- if there is a one-to-one correspondence between the members of two sets, doesn't that mean that the two sets have the same number of members? If not, then what is the sense and worth of a one-to-one correspondence in number theory and algebra? But if so, then my question stands: how can there be as many odd integers as there are integers? Forget the talk of fractions and ratios. If each span of ten integers has twice as many integers as odd integers, at what point does the number of odd integers in an interval begin to exceed the number of integers in that interval, so that the total of odd integers can be equal to (in one-to-one correspondence) with the integers?

Sorry. There ~has~ to be twice as many integers as odd integers. If it is so for any finite range of integers (granting it has an even number of integers), then it has to be so for an infinite range of integers. One-to-one correspondence notwithstanding.

REB

The correspondence is NOT useful in algebra. It is used in comparing infinite series to see if they converge or diverge. http://en.wikipedia.org/wiki/Convergent_series

You can't ask how there can be as many odds as there are integers and then say drop the talk of ratios - you have just asserted a ratio of one to one. The two series are not "actually equal." You can't say that the one is bigger than the other. You can't talk about finite intervals or at what point either, since you are dealing in infinities. You cannot use the word equal in the same sense as expressing 2+2=4, since that is a ratio.

[Dennis Hardin]

The error in the Black Swans example consists in the conclusion that color is an essential attribute somehow connected to the nature of Swans. That is an inductive hypothesis based on enumeration, not an inductive inference based on an analysis of the nature of Swans.

But it is a generalization from a set of particulars to a universal. It happens to be an erroneous generalization. The induction by enumeration permits the assumption that color is a consequence of all the other identifiable characteristics of crows, swans and such like. And THAT is a problem. Also, inductions can be incomplete. There is always the possibility that a contrary fact will be found later one which destroys the generalization. Induction is guaranteed to work only after one has every last possible fact, which is something that is not going to happen. Induction is a perfect method for omnipotent folks, But for ordinary mortals knowing only some of the possible facts (but not all) it is not guaranteed to work. Sometimes it does, sometimes it doesn't.

Ask yourself what error Newton made when he arrived at his law of gravitation based on observation.

The scientific errors you refer to can be attributed to simple human fallibility. The fact that scentists make mistakes does not invalidate the method of induction, which is, quite simply, the only way we have to acquire new knowledge.

Going to causes and their enumeration, Mills method works very well in the macroscopic domain. The Mill approach simply cannot cope with the way the physical world operates at the subatomic level.

Ba'al Chatzaf

I see no basis for concluding that Mill’s Methods will not work just as well on the subatomic level. As more evidence of microscopic activity becomes available through innovations in instrumentation, we should be able to isolate causal factors in the exact same way as we presently do in laboratory settings. The one issue that does get tricky is knowing when we have done an adequate job of looking at variable factors. But the fact that the philosophy of science needs some work in that area does not justify your conclusion that induction is therefore fundamentally invalid. That’s just foolishness, and you are obviously far too bright to be advocating any such notion.

…

[BaalChatzaf]

Sorry. There ~has~ to be twice as many integers as odd integers. If it is so for any finite range of integers (granting it has an even number of integers), then it has to be so for an infinite range of integers. One-to-one correspondence notwithstanding.

REB

Before I get to my main point what is the "ratio" between the count of the integers and the count of the square integers?

Now, suppose a kid has two piles of pebbles, one pile is white pebbles the other pile is black pebbles. If he can match up the piles up one for one then they have the same (cardinal) number of pebbles. The kid does not even have to know how to count. That is how babies who do not know (yet) how to count, can count.

Now continue the thinking. Is the set of integers finite or not? If it is finite, then there must be a largest integer. Call that integer L. Now consider L + 1. Oops! Conclusion; the set of integers is not finite. One of the characteristics of a set with an infinite (transfinite) cardinal number is that it can be put into one to one correspondence with a proper subset of itself. In the case of the integers the even integers (for example) is a proper subset of the integers. Why? There are integers which are not even so they are not in the set of even integers. But the set of even integers can be matched up one to one with the set of all integers, we have already done that. Ditto for the set of square integers (integers = n*n for some n). The set of square integers can be matched up one to one with the set of integers. Here is the mapping: n <-> n*n ( = n^2 ). By the way, Galileo noticed this last en passant, but he did not dwell on it.

How does the cardinality of the real numbers compare with the cardinality of the integers (or the rationals). It turns out the cardinality of the real numbers is large than the cardinality of the rationals (or the integers). Why? Because one can show there is no one to one correspondence between the set of integers and the set of reals. This is done by the famous Diagonal Proof of Cantor. If you can not accept the cardinality of the integers and the square integers then there is no way you are going to accept the Diagonal Proof.

See: http://en.wikipedia..../Diagonal_proof

If this does not convince you, than I have hit a wall with you. It is the same kind of thing when I run into Creationists or Flat-Earth people. But you are in good company. Georg Cantor who invented the theory of transfinite numbers ran into the same wall with some of the leading mathematicians of Europe*. But in the long run Cantor's ideas won out. As David Hilbert once said, at an address to mathematicians --- "From this Paradise which Cantor has created for us, no one will drive us out..."

See: http://en.wikipedia.org/wiki/Georg_Cantor

Ba'al Chatzaf

*

Cantor's theory of transfinite numbers was originally regarded as so counter-intuitive—even shocking—that it encountered resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré^[3] and later from Hermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections. Some Christian theologians (particularly neo-Scholastics) saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God,^[4] on one occasion equating the theory of transfinite numbers with pantheism.^[5] The objections to his work were occasionally fierce: Poincaré referred to Cantor's ideas as a "grave disease" infecting the discipline of mathematics,^[6] and Kronecker's public opposition and personal attacks included describing Cantor as a "scientific charlatan", a "renegade" and a "corrupter of youth."^[7] Writing decades after Cantor's death, Wittgenstein lamented that mathematics is "ridden through and through with the pernicious idioms of set theory," which he dismissed as "utter nonsense" that is "laughable" and "wrong".^[8] Cantor's recurring bouts of depression from 1884 to the end of his life were once blamed on the hostile attitude of many of his contemporaries,^[9] but these episodes can now be seen as probable manifestations of a bipolar disorder.

Edited September 13, 2010 by BaalChatzaf

[merjet]

Roger,

We disagreed about zero, but I am on your side here. Saying there as many even (or odd) integers as there are all integers, because a one-to-one correspondence is possible, doesn't fly. Start with a whole -- all the integers -- and remove half of them -- the odd (or even) ones. Saying there are as many after as before is nonsense and violates part-whole logic.

Here is another correspondence to support our side. For all integers x > 0, let y = x if x is even and y = x+1 if x is odd. The domain is all the positive integers and the function is a 2-to-1 correspondence, which implies there are twice as many elements in the domain as the range. One-to-one does not have a monopoly.

Edited September 13, 2010 by Merlin Jetton

[Roger Bissell]

Sorry. There ~has~ to be twice as many integers as odd integers. If it is so for any finite range of integers (granting it has an even number of integers), then it has to be so for an infinite range of integers. One-to-one correspondence notwithstanding.

REB

Before I get to my main point what is the "ratio" between the count of the integers and the count of the square integers?

Now, suppose a kid has two piles of pebbles, one pile is white pebbles the other pile is black pebbles. If he can match up the piles up one for one then they have the same (cardinal) number of pebbles. The kid does not even have to know how to count. That is how babies who do not know (yet) how to count, can count.

Now continue the thinking. Is the set of integers finite or not? If it is finite, then there must be a largest integer. Call that integer L. Now consider L + 1. Oops! Conclusion; the set of integers is not finite. One of the characteristics of a set with an infinite (transfinite) cardinal number is that it can be put into one to one correspondence with a proper subset of itself. In the case of the integers the even integers (for example) is a proper subset of the integers. Why? There are integers which are not even so they are not in the set of even integers. But the set of even integers can be matched up one to one with the set of all integers, we have already done that. Ditto for the set of square integers (integers = n*n for some n). The set of square integers can be matched up one to one with the set of integers. Here is the mapping: n <-> n*n ( = n^2 ). By the way, Galileo noticed this last en passant, but he did not dwell on it.

How does the cardinality of the real numbers compare with the cardinality of the integers (or the rationals). It turns out the cardinality of the real numbers is large than the cardinality of the rationals (or the integers). Why? Because one can show there is no one to one correspondence between the set of integers and the set of reals. This is done by the famous Diagonal Proof of Cantor. If you can not accept the cardinality of the integers and the square integers then there is no way you are going to accept the Diagonal Proof.

See: http://en.wikipedia..../Diagonal_proof

If this does not convince you, than I have hit a wall with you. It is the same kind of thing when I run into Creationists or Flat-Earth people. But you are in good company. Georg Cantor who invented the theory of transfinite numbers ran into the same wall with some of the leading mathematicians of Europe*. But in the long run Cantor's ideas won out. As David Hilbert once said, at an address to mathematicians --- "From this Paradise which Cantor has created for us, no one will drive us out..."

See: http://en.wikipedia.org/wiki/Georg_Cantor

Ba'al Chatzaf

*

Cantor's theory of transfinite numbers was originally regarded as so counter-intuitive—even shocking—that it encountered resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré^[3] and later from Hermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections. Some Christian theologians (particularly neo-Scholastics) saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God,^[4] on one occasion equating the theory of transfinite numbers with pantheism.^[5] The objections to his work were occasionally fierce: Poincaré referred to Cantor's ideas as a "grave disease" infecting the discipline of mathematics,^[6] and Kronecker's public opposition and personal attacks included describing Cantor as a "scientific charlatan", a "renegade" and a "corrupter of youth."^[7] Writing decades after Cantor's death, Wittgenstein lamented that mathematics is "ridden through and through with the pernicious idioms of set theory," which he dismissed as "utter nonsense" that is "laughable" and "wrong".^[8] Cantor's recurring bouts of depression from 1884 to the end of his life were once blamed on the hostile attitude of many of his contemporaries,^[9] but these episodes can now be seen as probable manifestations of a bipolar disorder.

Why am I not surprised about Cantor. :-/ Reminds me of the old saying, "Are you being paranoid if they really are against you?"

Seriously, regarding "infinite sets" such as the integers -- how can the concept of "set" even apply to such a..."collection"? By its nature, doesn't "collection" or "set" or "group" (in the everyday, not the mathematical sense) refer to something that is bounded, having a finite number of members? Isn't it illegitimate on the face of it to extend these concepts to the realm of the infinite?

REB

[Roger Bissell]

Roger,

We disagreed about zero, but I am on your side here. Saying there as many even (or odd) integers as there are all integers, because a one-to-one correspondence is possible, doesn't fly. Start with a whole -- all the integers -- and remove half of them -- the odd (or even) ones. Saying there are as many after as before is nonsense and violates part-whole logic.

Here is another correspondence to support our side. For all integers x > 0, let y = x if x is even and y = x+1 if x is odd. The domain is all the positive integers and the function is a 2-to-1 correspondence, which implies there are twice as many elements in the domain as the range. One-to-one does not have a monopoly.

Thanks, Merlin -- but Ted seems to be arguing (with Ba'al) that "twice as many" does not apply to the infinite domain any more than "ratio", and that "the same as," "as many as," and "equal" fall by the same sword. I am going to study this a bit more and see if I can't sharpen up my argument (Pat Corvini of ARI has some interesting things to say about this that I might crib for our discussion), but that's all I'm going to comment on it for now.

As for our old buddy, "zero," I want to look at that a bit more, too. I suspect that I took a wrong turn at Albuquerque, but I still like the idea that zero is an operation-stopper (in the sense of something that will not allow things like addition, subtraction, squaring, etc. to proceed). More on that another time, perhaps.

REB

[BaalChatzaf]

Why am I not surprised about Cantor. :-/ Reminds me of the old saying, "Are you being paranoid if they really are against you?"

Seriously, regarding "infinite sets" such as the integers -- how can the concept of "set" even apply to such a..."collection"? By its nature, doesn't "collection" or "set" or "group" (in the everyday, not the mathematical sense) refer to something that is bounded, having a finite number of members? Isn't it illegitimate on the face of it to extend these concepts to the realm of the infinite?

REB

An integer is any number I can get to by adding 1 to 0 one or more instances. This gives me

1, 2, 3... and so on. The collection of such numbers is the set of integers. Give me any number. In a finite sequence of steps I can determine whether it is an integer or not.

And no it is not illegitimate to extend indefinitely. Mathematicians have been doing it for over 100 years and have eliminated all known contradictions. Mathematics works.

Most of the major theorems in math have been proved using transfinite methods and with the axiom of choice (another bone of contention). Kronecker and his conservative objector colleagues have long since lost the fight. David Hilbert and his friends have won.

If one forbids transfinite reasoning and the axiom of choice over half the useful theorems in analysis will disappear. Say goodbye to topology, to limits, to proofs of uncomputability etc.. Andrew Wiles proof of Fermat's Last Theorem would simply not be.

The theory of sets had a rough start. First there was the Russel-Frege paradox (the set of all sets which are not members of themselves). Then there were other more subtle paradoxes. They have been eliminated by avoidance and apparently no more have been found. Thing of the avoidance rules as being analogous to fuses in an electrical circuit. They exist to prevent short circuits and consequential damage. Mathematics is alive and well and has never been as productive as it is now. There is an old saying: the best revenge is living well. Well the Mathematicians have had their revenge against the set theory objectors.

Ba'al Chatzaf