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.

Your first sentence parrots Cantor's hoax.

Other than Objectivists have rejected Cantor's ideas. I side with the eminent physicist and mathematician Henri Poincaré, who held that "most of the ideas of Cantorian set theory should be banished from mathematics once and for all" (link). Poincaré was dismayed by Georg Cantor's theory of transfinite numbers, and referred to it as a "disease" from which mathematics would eventually be cured (link).

David Hilbert thje greatest mathematician of his time said "From the Paradise created for us by Cantor, no one shall drive us out:"

Sounds like a fanatic, religious nut.

He was the greatest mathematician of his day (late 19th century early 20 th century). His geometrical work provides the basis of quantum physics (Hilbert Space). His work in logic and set theory is of great importance. He work in analysis is the basis of the modern theory of linear transformation. He also got to the Theory of General Relativity before Einstein, but he gave Einstein the credit because Einstein's approach was a hint he used to derive the gravitational field equations. In addition to being a front line mathematician he identified the 25 most important problems in mathematics that existed around 1913 and his list of problems drove much of the mathematical research that was done prior to 1950. In 1899 in his book Grundlagen der Geometrie he cleaned up Euclid's act. He got rid of all the gaps and defects in Euclid's Elements that had been hanging in there for nearly 2100 years.

Hilbert also made it possible for Emmy Neuter to have a university position in mathematics at a time when it was damned near impossible for women to break into the field. Emmy Neuter proved the famous theorem which matched group invariants to the conservation laws of physics. Einstein considered her contributions to physics as indispensable.

There is your "religious nut".

Your reaction to set theory is isomorphic to that of Leonard Peikoff. who is a mathematical ignoramus. I had this same argument with L.P. on WBZ radio (the David Brudnoy Show) about 35 years ago.

Been there. Done that. That is why I no longer strenuously argue the shape of the Earth with members of the Flat Earth Society.

Here is a summary of Hilbert's accomplishments, that "religious nut".

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.

Mis-speak. -A- way of comparing quantities is a 1-1 correspondence.

If you have another and it is logically consistent, then use it. As long as 2 + 2 = 4 it is cool.

-A- way of comparing quantities is a 1-1 correspondence.

If you have another and it is logically consistent, then use it. As long as 2 + 2 = 4 it is cool.

y = x if x even, x+1 if x odd. That is a 2-to-1 map/function/correspondence, which implies the set X (all integers) has twice as many numbers as the set 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.

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

Who said it was the -only- way. It is -a way-. If you define equi cardinality as the existence of a 1-1 onto mapping between sets then this relation has the three properties that any righteous equivalency has: it is reflexive, symmetry and transitive.

If you know another definition of equality between objects (please specify which objects) then pray do let us know what you have in mind.

As long as what you come up with leads to no contradictions and has the three properties of a righteous equivalence you are o.k.

Let us know what you had in mind.

By the way, if you restrict cardinal quantities to the cardinals of finite sets (no matter how) large you will not get the feature that a set can be put in 1-1 correspondence with a proper subset of itself. So if you want quantities that a proper Objectivist would not Object to, then restrict yourself to finite sets.

-A- way of comparing quantities is a 1-1 correspondence.

If you have another and it is logically consistent, then use it. As long as 2 + 2 = 4 it is cool.

y = x if x even, x+1 if x odd. That is a 2-to-1 map/function/correspondence, which implies the set X (all integers) has twice as many numbers as the set Y (even integers).

That is ill defined. What are two sets and what is the mapping.;

Let the two sets be A and B, both subsets of the integers.

Now let x in A. If x is even you say f(x) = x

and if x is odd you say f(x) = x + 1.

if x = x' then f(x) = f(x') now let us check the -inverse-

suppose f(x) = f(x') does this imply x = x'. No

Example f(4) = 4 but f(3) = 4 so the inverse function is not well defined.

So, 4 and 3 which are not equal get mapped into the same element. bzzzzzzzt. Wrong.

So your function as you defined it is NOT a 1-1 onto mapping. It has an ill defined inverse.

That is ill defined. What are two sets and what is the mapping.;

Let the two sets be A and B, both subsets of the integers.

Now let x in A. If x is even you say f(x) = x

and if x is odd you say f(x) = x + 1.

if x = x' then f(x) = f(x') now let us check the -inverse-

suppose f(x) = f(x') does this imply x = x'. No

Example f(4) = 4 but f(3) = 4 so the inverse function is not well defined.

So, 4 and 3 which are not equal get mapped into the same element. bzzzzzzzt. Wrong.

So your function as you defined it is NOT a 1-1 onto mapping. It has an ill defined inverse.

Wrong. Of course, it is not a 1-to-1 mapping. I clearly said it was a 2-to-1. What's next, f(x)=x^2 isn't well-defined because both x=2 and x=-2 map to the same element, 4?

Go back and try something else. Do something useful.

That is ill defined. What are two sets and what is the mapping.;

Let the two sets be A and B, both subsets of the integers.

Now let x in A. If x is even you say f(x) = x

and if x is odd you say f(x) = x + 1.

if x = x' then f(x) = f(x') now let us check the -inverse-

suppose f(x) = f(x') does this imply x = x'. No

Example f(4) = 4 but f(3) = 4 so the inverse function is not well defined.

So, 4 and 3 which are not equal get mapped into the same element. bzzzzzzzt. Wrong.

So your function as you defined it is NOT a 1-1 onto mapping. It has an ill defined inverse.

Wrong. Of course, it is not a 1-to-1 mapping. I clearly said it was a 2-to-1. What's next, f(x)=x^2 isn't well-defined because both x=2 and x=-2 map to the same element, 4?

Go back and try something else. Do something useful.

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## merjet

Your first sentence parrots Cantor's hoax.

Other than Objectivists have rejected Cantor's ideas. I side with the eminent physicist and mathematician Henri Poincaré, who held that "most of the ideas of Cantorian set theory should be banished from mathematics once and for all" (link). Poincaré was dismayed by Georg Cantor's theory of transfinite numbers, and referred to it as a "disease" from which mathematics would eventually be cured (link).

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## BaalChatzaf

Of Cantor's "hoax" David Hilbert thje greatest mathematician of his time said "From the Paradise created for us by Cantor, no one shall drive us out:"

Mathematics is doing just fine and it is most based on ZFC set theory.

But it does not surprise me that a member of the Flat Earth Society would curse our maps as a hoax.

Bob Kolker

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## merjet

Sounds like a fanatic, religious nut.

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## BaalChatzaf

He was the greatest mathematician of his day (late 19th century early 20 th century). His geometrical work provides the basis of quantum physics (Hilbert Space). His work in logic and set theory is of great importance. He work in analysis is the basis of the modern theory of linear transformation. He also got to the Theory of General Relativity before Einstein, but he gave Einstein the credit because Einstein's approach was a hint he used to derive the gravitational field equations. In addition to being a front line mathematician he identified the 25 most important problems in mathematics that existed around 1913 and his list of problems drove much of the mathematical research that was done prior to 1950. In 1899 in his book Grundlagen der Geometrie he cleaned up Euclid's act. He got rid of all the gaps and defects in Euclid's Elements that had been hanging in there for nearly 2100 years.

Hilbert also made it possible for Emmy Neuter to have a university position in mathematics at a time when it was damned near impossible for women to break into the field. Emmy Neuter proved the famous theorem which matched group invariants to the conservation laws of physics. Einstein considered her contributions to physics as indispensable.

There is your "religious nut".

Your reaction to set theory is isomorphic to that of Leonard Peikoff. who is a mathematical ignoramus. I had this same argument with L.P. on WBZ radio (the David Brudnoy Show) about 35 years ago.

Been there. Done that. That is why I no longer strenuously argue the shape of the Earth with members of the Flat Earth Society.

Here is a summary of Hilbert's accomplishments, that "religious nut".

http://en.wikipedia.org/wiki/David_Hilbert

Ba'al Chatzaf

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## merjet

I have never been a member of the Flat Earth Society.

Here is a summary of Henri Poincaré's accomplishments (link).

Still clinging to your faulty premise, eh?

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## BaalChatzaf

Both Poincare and Hilbert were top mathematicians in the late 19 th and early 20 th century.

Ba'al Chatzaf

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## BaalChatzaf

Mis-speak. -A- way of comparing quantities is a 1-1 correspondence.

If you have another and it is logically consistent, then use it. As long as 2 + 2 = 4 it is cool.

Ba'al Chatzaf

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## merjet

y = x if x even, x+1 if x odd. That is a 2-to-1 map/function/correspondence, which implies the set X (all integers) has twice as many numbers as the set Y (even integers).

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## BaalChatzaf

Who said it was the -only- way. It is -a way-. If you define equi cardinality as the existence of a 1-1 onto mapping between sets then this relation has the three properties that any righteous equivalency has: it is reflexive, symmetry and transitive.

If you know another definition of equality between objects (please specify which objects) then pray do let us know what you have in mind.

As long as what you come up with leads to no contradictions and has the three properties of a righteous equivalence you are o.k.

Let us know what you had in mind.

By the way, if you restrict cardinal quantities to the cardinals of finite sets (no matter how) large you will not get the feature that a set can be put in 1-1 correspondence with a proper subset of itself. So if you want quantities that a proper Objectivist would not Object to, then restrict yourself to finite sets.

Ba'al Chatzaf

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## BaalChatzaf

That is ill defined. What are two sets and what is the mapping.;

Let the two sets be A and B, both subsets of the integers.

Now let x in A. If x is even you say f(x) = x

and if x is odd you say f(x) = x + 1.

if x = x' then f(x) = f(x') now let us check the -inverse-

suppose f(x) = f(x') does this imply x = x'. No

Example f(4) = 4 but f(3) = 4 so the inverse function is not well defined.

So, 4 and 3 which are not equal get mapped into the same element. bzzzzzzzt. Wrong.

So your function as you defined it is NOT a 1-1 onto mapping. It has an ill defined inverse.

Go back and try something else.

or take a course in algebra or set theory.

Do something useful.

Ba'al Chatzaf

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## merjet

Wrong. Of course, it is not a 1-to-1 mapping. I clearly said it was a 2-to-1. What's next, f(x)=x^2 isn't well-defined because both x=2 and x=-2 map to the same element, 4?

Go back and try something else. Do something useful.

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## BaalChatzaf

Why not a 3 to

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