Para-consistent logic

RBerenguel · **#41**

SmoothOper wrote:

Bill Spight wrote:

SmoothOper wrote:

Actually, I didn't read that carefully enough. I think it is really annoying when mathematicians fail to work at set level. For example everyone knows that the integers I are a subset of the rationals R therefore there must be more rationals than the integers.

As is often the case, what "everybody knows" is not so.

For one thing, integers are not a subset of rationals, strictly speaking. A rational is an ordered pair of integers. True, there are rationals that are equal to integers, but that is not exactly the same thing.

Quote:

However there is a cockamamie proof where mathematicians make a mapping from integers to IxI then claim that there are as many I as R.

What does it mean to say that two infinities are equal, or that one infinity is greater than another? If you do not admit absolute infinities, then the question is meaningless. Fine. But then you do not get to say that there are more rationals than integers.

Someone doesn't understand the axiom of determinancy. 1:2 is not a rational number (1,2) is not a rational number, 1/2 is a rational number given the operation of division and equivalency classes, however if you have a suitable set of equivalency classes, then you can see that there are more rationals than integers, since integers are a subset. If you don't have the equivalency classes you can't define division, thus your sets of tuples aren't rational numbers. Some people who believe in the axiom of choice will sort of choose to do the mapping implicitly to make the proof work, other people won't be so kind as to go along.

Technically, the equivalence classes used to define rationals don't (strictly) depend on division. They are just equivalence classes, which happen to behave pretty much like what a layman thinks is a rational number. It's like defining tensors as universal objects in categories. The fact that they can afterwards be used for something does not come for granted, or should imply categories behave just so.

Edit: things->thinks

Tryss · **#42**

SmoothOper wrote:

Some people who believe in the axiom of choice will sort of choose to do the mapping implicitly to make the proof work, other people won't be so kind as to go along.

But we can make the mapping explicitly :

And this is indeed a bijection (you can make all the steps backward) :

Obviously, you need to have proof that the decomposition in prime factors of a positive exist and is unique.

SmoothOper · **#43**

RBerenguel wrote:

Technically, the equivalence classes used to define rationals don't (strictly) depend on division. They are just equivalence classes, which happen to behave pretty much like what a layman thinks is a rational number. It's like defining tensors as universal objects in categories. The fact that they can afterwards be used for something does not come for granted, or should imply categories behave just so.

Edit: things->thinks

However, division does depend on rational equivalency classes, and isn't defined over integers.

SmoothOper · **#44**

Tryss wrote:

22/7

So 22 is a rational and 7 is a rational and division is an operation defined on the rationals 22 and 7, 22 happens to also be an integer, but the result of twenty two divided by seven is not an integer, therefore there are more rationals than integers.

Bill Spight · **#45**

SmoothOper wrote:

Bill Spight wrote:

SmoothOper wrote:

Actually, I didn't read that carefully enough. I think it is really annoying when mathematicians fail to work at set level. For example everyone knows that the integers I are a subset of the rationals R therefore there must be more rationals than the integers.

As is often the case, what "everybody knows" is not so.

For one thing, integers are not a subset of rationals, strictly speaking. A rational is an ordered pair of integers. True, there are rationals that are equal to integers, but that is not exactly the same thing.

Quote:

However there is a cockamamie proof where mathematicians make a mapping from integers to IxI then claim that there are as many I as R.

What does it mean to say that two infinities are equal, or that one infinity is greater than another? If you do not admit absolute infinities, then the question is meaningless. Fine. But then you do not get to say that there are more rationals than integers.

Someone doesn't understand the axiom of determinancy. 1:2 is not a rational number (1,2) is not a rational number, 1/2 is a rational number given the operation of division and equivalency classes, however if you have a suitable set of equivalency classes, then you can see that there are more rationals than integers, since integers are a subset.

I did not say that ordered pairs of integers are rationals, I said the converse. Does someone understand English?

Given two integers, A and B, with B != 0, we may define a rational number as an ordered pair, (A,B), with certain properties. For instance, (A,B) = (C,D) iff A*D = B*C. And (A,B)*(C,D) = (A*C,B*D). And so on.

Bill Spight · **#46**

SmoothOper wrote:

So 22 is a rational and 7 is a rational and division is an operation defined on the rationals 22 and 7, 22 happens to also be an integer, but the result of twenty two divided by seven is not an integer, therefore there are more rationals than integers.

So dividing one integer by another produces something that is not an integer?
Sorry, I have it on good authority that division is not defined over integers.

SmoothOper wrote:

However, division does depend on rational equivalency classes, and isn't defined over integers.

RBerenguel · **#47**

SmoothOper wrote:

RBerenguel wrote:

Technically, the equivalence classes used to define rationals don't (strictly) depend on division. They are just equivalence classes, which happen to behave pretty much like what a layman thinks is a rational number. It's like defining tensors as universal objects in categories. The fact that they can afterwards be used for something does not come for granted, or should imply categories behave just so.

Edit: things->thinks

However, division does depend on rational equivalency classes, and isn't defined over integers.

Huh? So what?

Bill Spight · **#48**

OK. So division is not defined over integers, which means that we cannot define a rational number as the result of division of integers. However, we can define division for rational numbers.

Given rational numbers, (A,B) and (C,D), if C!=0 then (A,B)/(C,D) = (A*D,B*C).

lemmata · **#49**

SmoothOper wrote:

lemmata wrote:

Pippen wrote:

I should mention I am very skeptic of infinities also. I do not think we can prove them, we can just prove that "it goes on and on and on and on [but we do not know if there is an end in fantasiciollion years]". I find it paradoxical and inconsistent to talk about infinite sets, because since this set has infinite objects it is never finished and stable. Every proof about this set has to be incomplete.

Also, modern higher math uses variables to prove things about infinite sets. E.g. they prove that there are infinite natural numbers, because every number n has a successor n+1, so that there can't be a last one. BUT: That assumes that "n" stands for all possbile natural numbers, infinitely many as we just saw. How can one assume that? How can a single letter stand for 1. a single number but 2. at the same time for all? And on top of that there are no rules/axioms about that, it's just pure assumption and practice.

Therefore I like the "only what we can acutally calculate (even with a computer)"-math. Anything else is metaphysics in disguise.

People usually do not object to statements like "each child has a biological father" by proclaiming that there is no way that the word "child" can stand for every child on earth. This is essentially the objection you are raising about the statement in the form "every child c has a biological father f". However, latter is just the mathematical style of expressing the former.

Actually, I didn't read that carefully enough. I think it is really annoying when mathematicians fail to work at set level. For example everyone knows that the integers I are a subset of the rationals R therefore there must be more rationals than the integers. However there is a cockamamie proof where mathematicians make a mapping from integers to IxI then claim that there are as many I as R. Though in my opinion, it is impossible to do division on all integers without a mapping to the rationals R in which the integers I are a strict subset of the rationals. Though I could see how you could go for that kind of proof if you don't have a concept where you can't refer to an infinite set of things.

Please do not talk about what you do not understand with such authority.

You are incorrect about the set theoretic notion of size, which is equinumerosity. Two sets are of equal size if and only if there is a one-to-one and onto mapping (a relabelling "set isomorphism") between them. This is like Noah's Ark. How do we know that there are an equal number of female and male animals on the Ark? Each female animal can be paired with a male animal in a way that leaves no animal unpaired. Every rational number can be paired with an integer in a way that leaves no rational or integer unpaired. This idea of pairing via set isomorphism is the set theoretic notion of size.

Please stop spreading bullshit about mathematics. There is no such thing as "opinion" on the facts of mathematics. If you have an "opinion" on such matters, that's probably a good sign that you don't know jack about it.

Signed,

Professional Mathematician

PS: Nearly all of mathematics is done with sets. It's hidden beneath the notation but well-founded. Even the integers are constructed using set theory axioms.

SmoothOper · **#50**

lemmata wrote:

You are incorrect about the set theoretic notion of size, which is equinumerosity. Two sets are of equal size if and only if there is a one-to-one and onto mapping (a relabelling "set isomorphism") between them. This is like Noah's Ark. How do we know that there are an equal number of female and male animals on the Ark? Each female animal can be paired with a male animal in a way that leaves no animal unpaired. Every rational number can be paired with an integer in a way that leaves no rational or integer unpaired. This idea of pairing via set isomorphism is the set theoretic notion of size.

Please stop spreading bullshit about mathematics. There is no such thing as "opinion" on the facts of mathematics. If you have an "opinion" on such matters, that's probably a good sign that you don't know jack about it.

Signed,

Professional Mathematician

PS: Nearly all of mathematics is done with sets. It's hidden beneath the notation but well-founded. Even the integers are constructed using set theory axioms.

Ah look at the cute mathematician he talks with such authority :salute:

. Ah, he doesn't like what I said, it makes him upset, he thinks it is my opinion. wah wah :roll:

Now lets watch the mathematician keep counting the pairs of things that aren't rational numbers. 1,2,3,4.... I know lets take one of the animals on the ark and cut it in half, that would be rational now wouldn't it. Giraffe/Lion=Cow

RBerenguel · **#51**

SmoothOper wrote:

lemmata wrote:

You are incorrect about the set theoretic notion of size, which is equinumerosity. Two sets are of equal size if and only if there is a one-to-one and onto mapping (a relabelling "set isomorphism") between them. This is like Noah's Ark. How do we know that there are an equal number of female and male animals on the Ark? Each female animal can be paired with a male animal in a way that leaves no animal unpaired. Every rational number can be paired with an integer in a way that leaves no rational or integer unpaired. This idea of pairing via set isomorphism is the set theoretic notion of size.

Please stop spreading bullshit about mathematics. There is no such thing as "opinion" on the facts of mathematics. If you have an "opinion" on such matters, that's probably a good sign that you don't know jack about it.

Signed,

Professional Mathematician

PS: Nearly all of mathematics is done with sets. It's hidden beneath the notation but well-founded. Even the integers are constructed using set theory axioms.

Ah look at the cute mathematician he talks with such authority :salute:

. Ah, he doesn't like what I said, it makes him upset, he thinks it is my opinion. wah wah :roll:

Now lets watch the mathematician keep counting the pairs of things that aren't rational numbers. 1,2,3,4.... I know lets take one of the animals on the ark and cut it in half, that would be rational now wouldn't it. Giraffe/Lion=Cow

It's not "liking". What you said is wrong. At the high mathematics level, stupid.

SmoothOper · **#52**

RBerenguel wrote:

SmoothOper wrote:

lemmata wrote:

You are incorrect about the set theoretic notion of size, which is equinumerosity. Two sets are of equal size if and only if there is a one-to-one and onto mapping (a relabelling "set isomorphism") between them. This is like Noah's Ark. How do we know that there are an equal number of female and male animals on the Ark? Each female animal can be paired with a male animal in a way that leaves no animal unpaired. Every rational number can be paired with an integer in a way that leaves no rational or integer unpaired. This idea of pairing via set isomorphism is the set theoretic notion of size.

Please stop spreading bullshit about mathematics. There is no such thing as "opinion" on the facts of mathematics. If you have an "opinion" on such matters, that's probably a good sign that you don't know jack about it.

Signed,

Professional Mathematician

PS: Nearly all of mathematics is done with sets. It's hidden beneath the notation but well-founded. Even the integers are constructed using set theory axioms.

Ah look at the cute mathematician he talks with such authority :salute:

. Ah, he doesn't like what I said, it makes him upset, he thinks it is my opinion. wah wah :roll:

Now lets watch the mathematician keep counting the pairs of things that aren't rational numbers. 1,2,3,4.... I know lets take one of the animals on the ark and cut it in half, that would be rational now wouldn't it. Giraffe/Lion=Cow

It's not "liking". What you said is wrong. At the high mathematics level, stupid.

Sure, I haven't bought into certain aspects of some peoples mathematics and set theory(mathematics and set theory that some may believe is smart). However, I have laid out good reasons to not believe those proofs and concepts, but that isn't any reason to get upset. I mean unless, you think a tantrum :oops:

might make people think you have a little extra :ugeek:

authority, which I assure you there aren't many professionals that behave that way though it is kind of fun to witness first hand :twisted:

.

leichtloeslich · **#53**

lemmata wrote:

You are incorrect about the set theoretic notion of size, which is equinumerosity. Two sets are of equal size if and only if there is a one-to-one and onto mapping (a relabelling "set isomorphism") between them.

While I firmly believe that SmoothOper is trolling, let's not forget that cardinality isn't the only possible measure of "size" for infinite sets. (Although it's arguably better than the partial order given by inclusion.)

Clearly there are more rationals than integers in an intuitive sense: the rationals are dense within the reals, while the integers are discrete.

Also there are clearly less primes than there are integers in general (in the sense of the prime number theorem, for example).

As for an example outside of \aleph_0, consider the cantor set: an uncountable subset of the reals with Lebesgue measure 0.

lemmata · **#54**

leichtloeslich wrote:

Clearly there are more rationals than integers in an intuitive sense: the rationals are dense within the reals, while the integers are discrete.

I only object to the underlined parts. No, it is not clearly true in an intuitive sense. It is absolutely true in a specific topological sense that is explicitly defined. In fact, the notion of dense sets is a topological one and requires more structure than the set theoretic notion of size. Well-founded mathematics rejects intuitive clarity as a valid method of proof. Many things that seem like they should be true turn out to be false. Intuition is very fallible when it comes to mathematical truth.

leichtloeslich wrote:

As for an example outside of \aleph_0, consider the cantor set: an uncountable subset of the reals with Lebesgue measure 0.

Again, the Lebesgue measure is generated by a particular topology. Furthermore, a measure theoretic notion of size is inappropriate for discussing the infinity of the rationals and integers. Both sets have Lebesgue measure zero---the same as the empty set---despite being infinite sets. The Lebesgue measure or the subset ordering are appropriate for other settings. I don't object to the existence of other notions of size. However: Right tools for the right problems.

leichtloeslich wrote:

While I firmly believe that SmoothOper is trolling, let's not forget that cardinality isn't the only possible measure of "size" for infinite sets. (Although it's arguably better than the partial order given by inclusion.)

That's why I usually try not to post in his threads. If I do, I try not to address him directly, but I have made an exception because a man can only take so much bs before he breaks.

SmoothOper wrote:

Ah look at the cute mathematician he talks with such authority . Ah, he doesn't like what I said, it makes him upset, he thinks it is my opinion. wah wah Now lets watch the mathematician keep counting the pairs of things that aren't rational numbers. 1,2,3,4.... I know lets take one of the animals on the ark and cut it in half, that would be rational now wouldn't it. Giraffe/Lion=Cow

My post wasn't for you. It was for the sane people who are interested in having conversations, not some troll trying to pretend to be a machine failing the Turing test. Your random sequences of non sequiturs about the conversation are fooling no one.

Mike Novack · **#55**

SmoothOper ...... I am not a professional mathematician, nor was my degree in mathematics (though decades ago I did teach math as well as the sciences as the secondary school level).

I think you are perhaps honest but confused, that you have seen certain "objections" but lack the background to understand what those do or do not mean. Your initial objection to the proof extending to infinity was specific and I tried to give you a reference to that area of math that doesn't use "induction".

But I really think that if you want to talk intelligently about these subjects you are going to have to study a great deal more math, standard math first, before trying to understand subsets of mathematics which aren't using certain axioms.

The problem you are having is with the notion of "truth", how this does or does not apply to mathematical statements. Essentially you are back before the latter part of the 19th Century when the crisis arose in mathematics (in this case geometry) over the "parallel postulate". At this point mathematicians realized that there could, be more than one "true" geometry (each depending in the axioms chosen) and the search began for the minimum set of axioms on which "math" was dependent. For example, reducing number theory to the Peano axioms (one of which is that one about induction).

It isn't correct for you to say "not true" (not a valid proof) when what you really mean is that you don't accept one of the axioms and you are only interested in what can be proven without it.

PS: Perhaps you might see what is going on with what bothers you about the integers and rationals by looking at the Euclidean geometery proof that a short line segment contains the same number of points as a longer one. There too you might have begum by marking off the length of the short one on the long one and saying "see, all the points inside that boundary pair up, so those beyond must be more."

But there is a different way. Join the two ends of each with a line. Those two lines meet in a point. From that point draw a line through any point on the short segment. It will also pass through a point on the longer segment. The postulates/axioms "two points determine a line" (only one line joins them) and "two lines determine a point" (only one intersection) will let you show that for every point in the longer segment there is one in the shorter segment << see what you were violating if you proposed that there were two points on the longer segment that shared the same point on the shorter segment >>.

What I am pointing out by this is to remind you that dealing with infinites and proofs by pairing are not something out of modern math.

hyperpape · **#56**

Bill Spight wrote:

For one thing, integers are not a subset of rationals, strictly speaking. A rational is an ordered pair of integers. True, there are rationals that are equal to integers, but that is not exactly the same thing.

Have you read Benacerraf's "What Numbers Could Not Be"? I don't have settled views here, but I have enough sympathy towards the mathematical structuralist view that I get a funny feeling when I hear that rationals are really just ordered pairs of integers.

Related: do you have an opinion about what the reals really are?

RBerenguel · **#57**

hyperpape wrote:

Bill Spight wrote:

For one thing, integers are not a subset of rationals, strictly speaking. A rational is an ordered pair of integers. True, there are rationals that are equal to integers, but that is not exactly the same thing.

Have you read Benacerraf's "What Numbers Could Not Be"? I don't have settled views here, but I have enough sympathy towards the mathematical structuralist view that I get a funny feeling when I hear that rationals are really just ordered pairs of integers.

Related: do you have an opinion about what the reals really are?

The reals are a really weird beast, born out of Cauchy's wedding with an epsilon

SmoothOper · **#58**

Fundamentally, as a computer scientist, I find it really annoying when Mathematicians play fast and loose with sets, then you try to implement their calculations and algorithms, and they don't work because they used the axiom of choice. Think compiler error.

RBerenguel · **#59**

SmoothOper wrote:

Fundamentally, as a computer scientist, I find it really annoying when Mathematicians play fast and loose with sets, then you try to implement their calculations and algorithms, and they don't work because they used the axiom of choice. Think compiler error.

Sure, happens all the time. Like when computationally determining bases for Banch spaces.

Bill Spight · **#60**

SmoothOper wrote:

Fundamentally, as a computer scientist, I find it really annoying when Mathematicians play fast and loose with sets, then you try to implement their calculations and algorithms, and they don't work because they used the axiom of choice. Think compiler error.

Having proved that you are not a mathematician, you now prove that you are not a computer scientist.

Para-consistent logic

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