Time well spent would be to learn some mathematics rather than claiming untrue things about it.

Anyway, did someone say "beyond infinity"? 'Beyond infinity' has no apparent meaning. First, again, 'infinity' should not be used as a noun in this context. Set theory does not define 'infinity'. Second, the set theoretic fact is that for any infinite cardinality there is a greater infinite cardinality. So, yes, in that sense, there is an infinite cardinality "beyond" the cardinality of the set of natural numbers, but there is no "beyond infinity".

Then why do you think bijection requires counting?



A quick look will tell you that there are twice as many feet as there are people. You do not need to count the number of people to know this to be true; just check for amputees...


Bijecting two feet for each person.

for any infinite cardinality there is a greater infinite cardinality — TonesInDeepFreeze

Which is the equivalent of saying beyond the quantity of infinity, there is a greater quantity of infinity (which is contradictory to the semantic of infinity). Again, you can add one to any quantity except of course the quantity of infinity.

Time well spent would be to learn some mathematics rather than claiming untrue things about it. — TonesInDeepFreeze

I don't believe I'm the one saying untrue things.

Then you believe an untruth.

And for about the fifth time, there is no object named by 'infinity'. So there is no object named by 'the quantity of infinity'.

There is least infinite cardinal, which is the cardinality of the set of the infinite set of natural numbers. And there are cardinals greater than the least infinite cardinal. Moreover, for each cardinal, whether it is a finite cardinal or infinite cardinal, there is a greater cardinal.

And when someone says "contradictory to the semantic of infinity" here it indicates that one did not read, or chose to ignore, what was written about that.

I'll repeat: Study of set theory is not a commitment to adherence to all the many meanings of 'infinite' and 'infinity' in everyday discourse, in philosophy and even other science. Rather, the adjective 'is infinite' has a special and very particular definition in set theory. It some ways it is compatible to other non set theoretic meanings of notions, but is not compatible with certain other non set theoretic meanings or notions.

No one disputes that the set theoretical definition might not accord with anyone's other notions. But that does make set theory inconsistent. A theory is inconsistent if and only if it proves some sentence P and not-P.

"There is a cardinality greater than the cardinality of the set of natural numbers" and your notion "There is no cardinality greater than the cardinality of the set of natural numbers" is not a contradiction of set theory, because set theory does not prove "There is no cardinality greater than the cardinality of the set of natural numbers" despite that that sentence is one you believe in your non set theoretic notions.

Banno humor...

A quick look will tell you that there are twice as many feet as there are people. You do not need to count the number of people to know this to be true; just check for amputees... — Banno

The benefit to me of what you've posted here is that I now reject the following from the OP and would change the last part of it in the link I provided to my post:

Perhaps one might argue that there is no count involved with regards to the latter and that it's just a fact that Infinity encompasses an infinite number of natural numbers. But if that's the case, then Infinity also encompasses an infinite number of possible real numbers and possible letters or possible x. But where there is no counting involved, all infinites are of the same size/quantity (or rather, infinity is one quantity as opposed to different quantities). — Philosopher19

I still hold the belief that saying 1,2,3,4 ad infinitum or {1,2,3,4,...} does not mean one has shown an infinite number of natural numbers. One has essentially suggested a number sequence goes on forever. But since one cannot count to infinity, it is the case that the total number of items in that sequence will not be infinity. If I do not do this, I will hit contradictions. If I do this, I will avoid contradictions.

Seeing as your post benefited me, I should thank you. So thank you.

Anyway, did someone say "beyond infinity"? — TonesInDeepFreeze

To Infinity and Beyond!

I don't believe I'm the one saying untrue things — Philosopher19

Hey, no problem. Start with a definition of "untrue".

It seems to me that you think I'm not paying attention to what you're saying and I think you're not paying attention to what I'm saying. I think we should end our discussion.

Peace

My position is now probably best represented by the following:

The only reason something like a sequence of numbers can go on forever, is because of Infinity. It is not because the sequence of numbers are Infinite.

Two different things can go on forever at different speeds, but this does not mean that one will go farther than the other when both are set to go on forever. It may look that way if you were to try and "map the distance covered by one to the other", but neither will ever cover an Infinite amount of distance for one to be able to conclude something like "this Infinite distance covered is greater than that Infinite distance covered". Of course, this is not the same as saying something like "this amount of distance covered in Infinity is greater than that amount of distance covered in Infinity".

I am paying close attention to what you are saying. While I just now showed specifically and fundamentally how you skipped what I said.

And if you want to discontinue your posting in this thread, then you can discontinue it. No one is stopping you.

Another misconception:

"saying 1,2,3,4 ad infinitum or {1,2,3,4,...} does not mean one has shown an infinite number of natural numbers."

Yes, saying that for every natural number there is a greater natural number does not in and of itself imply that there is an infinite set of natural numbers. But {0 1 2 3 ...} is not notation that for every natural number there is a greater natural number, but rather it is an informal notation to stand for the set of all and only the natural numbers.

The way set theory proves there exists a set with all and only the natural numbers is by an axiom from which we prove that there exists a set with all and only the natural numbers.

So the objection that we can't extrapolate from "for every natural number there is a greater natural number" to "there is a set with all the natural numbers" is true but a huge strawman since set theory does NOT claim that we can extrapolate that way.

Again, such objections are a product of sheer unfamiliarity with the subject matter.

It is by tolerance that there are forums such as this that allow such spreading of confusion.

There is least infinite cardinal, which is the cardinality of the set of the infinite set of natural numbers. And there are cardinals greater than the least infinite cardinal. Moreover, for each cardinal, whether it is a finite cardinal or infinite cardinal, there is a greater cardinal. — TonesInDeepFreeze

We are in disagreement right there. You say you can have an infinite number of natural numbers. I say this statement will lead to a contradiction. That contradiction being "this infinite set is bigger than that infinite set". This is a contradiction because infinity is that which you cannot add to or have more than of. If you don't believe in this then how can we possibly agree?

This is why I said:

The only reason something like a sequence of numbers can go on forever, is because of Infinity. It is not because the sequence of numbers are Infinite. — Philosopher19

But {0 1 2 3 ...} is not notation that for every natural number there is a greater natural number, but rather it is an informal notation to stand for the set of all and only the natural numbers. — TonesInDeepFreeze

I don't deny that there is such a set, but I deny that the total number of natural numbers in this set reaches infinity. Imagine you have all the natural numbers in {1,2,3,4,...}. Can you show what number comes before infinity to be able to meaningfully assert something like {1,2,3,4,...} consists of an infinite number of natural numbers?

You can't just say it has all the numbers and all the numbers amount to infinity

Could it be your model of infinities is based on a mathematical concept and Philosopher19's model is based on a different mathematical concept and both are correct within their own frameworks. In a general sense I think that is very much in the realm of possibility and as a matter of good practice you shouldn't discount a different concept because it differs from your own standard methods.

An example:

Banno takes infinity minus one million and gets infinity.

You say you can't subtract from infinity.

I say an infinite set of integers minus the first million integers is a set with the first million integers removed and I could list them.

1, 2, 3, 4......

Philospher19, again ignored for the second time what I wrote about "contradiction".

I'd say pot calling the kettle black.

Mark Nyquist:

Now for the third time:

Anyone can have whatever concept of mathematics they want to have. But having a different concept from set theory doesn't entail that set theory itself proves any contradictions.

As to subtraction with infinite cardinals, again I say, just start by defining it.

And you skipped what I said about removing.

I also want to add one more thing to the following

But {0 1 2 3 ...} is not notation that for every natural number there is a greater natural number, but rather it is an informal notation to stand for the set of all and only the natural numbers. — TonesInDeepFreeze

Suppose something goes on forever such that it covers more and more distance as it goes on. So it covers 5km, 10km, 15km ad infinitum. I can't say {5km, 15km, 20km, ...km} is an informal notation to stand for all the distance it covered and that that distance is infinite. Do you see what I'm saying? You can't just say {1,2,3,4,...} is an informal notation to stand for the set of all and only the natural numbers and that the total number of natural numbers in that set is infinity.

I believe in the same way that I can't say the total distance covered is infinite, you can't say the total number of natural numbers in that set is infinity.

I'm no pot, since I haven't ignored what the poster said.

But you are such a pot, as recently you ignored what I said.

I am addressing your point. I believe you are not reading all of it. See my last post to you.

I can say it is an informal notation for the set of all and only the natural numbers because that is exactly what it is an informal notation for. The question of how a notation is used is settled empirically, by just looking at the way it is used. And in mathematics, when '{0 1 2 3 ...}' is used, it is used to stand for the set of all and only the set of natural numbers.

A separate question is whether in mathematics there exists such a set. And I addressed that, but you skipped what I wrote.

I said exactly what points you are not addressing, and now you just come back to insist that you are addressing them though you are not. And you said, "I believe you are not reading all of it. See my last post to you" before I even had a chance to reply to it, as then I did reply to it. How absurd to charge someone with not reading a post before he has even had a chance to reply to it in the order it was posted. Your posting is an absurd loop.

I don't know what more to say. When I use the label/word "infinity", I'm not sure you're focused on the same semantic that I'm focused on.

Your posting is an absurd loop — TonesInDeepFreeze

I'm sorry if discussing with me has been a negative experience for you.

For about the sixth time, and this is one of the points you keep refusing to address:

I don't begrudge anyone from having whatever concept and definition of infinitude they wish to have.

But having a different concept and definition of infinitude doesn't thereby entail that there is a contradiction in set theory or mathematics.

Again, yes, there may be a contradiction between set theory and certain other formulations. But that does not entail that there is a contradiction within set theory.

Again, for emphasis yet again, since you keep skipping this point, no one should deny you from having whatever concepts and definitions you would like to have, and if thereby set theory does not suit you or does not make sense to you, then so be it, but that doesn't entail that set theory leads to any contradiction in itself.

Yes, set theory does not have the same concept of infinitude that you have. As well as, which you also keep skipping, set theory does not refer to an object named 'infinity' but rather to the property of being infinite, which is a crucial distinction.

In this thread, there was discussion about set theory and that discussion had important errors. So I provided a systematic synopsis of the area in discussion as that synopsis corrected the errors and explained why they are in error. Then replies came to my posts, but certain of those replies still had misconceptions about set theory.

Again, espouse whatever concept of infinitude you wish. But that does not justify an incorrect and misinformed critique of set theory.

And whether the thread is or is not a negative experience for anyone, it still stands that your posting has been an absurd loop.

One might argue that the latter encompasses imagining that the count to infinity is complete, but one cannot imagine such a thing. — Philosopher19

The answer to your problem is quite simple. In mathematics things are done by axiom. If you want to count to infinity and beyond, simply produce an axiom which allows you to do that, and bingo the infinite is countable, and you're ready to go beyond. Look closely at the following:

The way set theory proves there exists a set with all and only the natural numbers is by an axiom from which we prove that there exists a set with all and only the natural numbers. — TonesInDeepFreeze

Then why do you think bijection requires counting? — Banno

Actually, the inverse is what is the case, counting is a form of bijection. But this does not necessarily imply that all bijections are a form of counting. And, some might still argue that there are forms of counting which wouldn't qualify as bijections. It all depends on how one might restrict these concepts though definition.

Note that 'countable' in mathematics does not mean that any human being can count every member of the set. Rather, 'countable' in mathematics only means that the set has a one-to-one correspondence with a natural number or the set of natural numbers.

Also, AGAIN, there is no object in set theory called 'the infinite'.

And it is not the case that set theory says that every infinite set is countable.

And a bijection is a certain kind of function. And set theory doesn't have a term 'counting' so set theory does not say anything about whether bijections are a form of counting.

Hmm. The difference between injective and bijective functions is more complex than I had thought. This Maths Is Fun site sets it out pretty clearly, and talks about ONTO. Cheers. I think my two feet for each head is bijective... feet = 2(heads), for positive integers...?

It's not complicated.

Definitions:

f is an injection iff f is a one-to-one function. We may also say 'f is an injective function'.

f is an injection from x into y iff (f is an injection and the domain of f is x and the range of f is a subset of y).

f is surjection from x onto y iff (f is a function and the domain of f is x and the range of f is y).

(So every function is a surjection from its domain onto its range.)

f is a bijection from x onto y iff (f is an injection and f is a surjection from x onto y).

(So every injection is a bijection from its domain onto its range.)

/

It seems to me that your feet-head is just an example of multiplication. Two feet for each of n number of people is 2*n feet.

Or it could be a bijection between the number of pairs of feet and number of people. There are the same number of pairs of feet as there is the number of people.

A clearer example would be just two lines of people such that we could see that there is a one-to-one correspondence between the people in one line and the people in the other line.

When considering a sequence of numbers, such as those between 0 and 4, where there are an infinite number of elements (including fractions and irrationals) between 0 and 1 and every other natural number in the sequence, the set of numbers between 0 and 2 would contain more elements than between 0 and 1. If one were to continuously move through this sequence of numbers, they would never reach the number 1 since one would have to move through an infinity of fractions first. In contrast, if the infinite sequence consists simply of natural numbers, reaching 1 would happen almost instantly.

This comparison illustrates that the "distance" covered between 0 and 1 in the first example is different from the "distance" between 0 and 1 in the last example. It can be concluded that the first example which includes the rationals and irrationals represents a larger infinity than simply an infinite series of natural numbers.

Does this explanation make sense? I admit that I haven't spent much time studying the intricacies of infinities and am not completely familiar with the technical terms and notations that mathematicians typically use to discuss these concepts.

To put your musings in perspective, here are the mathematical facts:

The set of rational numbers between any two natural numbers is not sequenced by the ordinary less-than relation on rational numbers.

Between any two natural numbers there is a denumerable sequence of the set of rational numbers between the natural numbers, but it is not isomorphic with the ordinary less-than relation on rational numbers.

The set of irrational numbers between any two natural numbers is not sequenced by any countable ordinal.

The set of irrational numbers between any two natural numbers is sequenced by some uncountable ordinal if we have the axiom of choice.

The set of rational numbers between any two natural numbers is equinumerous with the set of all natural numbers.

The set of irrational numbers between any two natural numbers is not equinumerous with the set of natural numbers.

It is not the case that there are more rational numbers between 0 and 2 than between 0 and 1.

It is not the case that there are more irrational numbers between 0 and 2 than between 0 and 1.

The set of rational numbers is equinumerous with the set of natural numbers.

Any infinite subset of the set of rational numbers is equinumerous with the set of natural numbers.

Any infinite subset of the set of rational numbers is equinumerous with the set of rational numbers.

The set of irrational numbers is not equinumerous with the set of natural numbers.

The set of irrational numbers is not equinumerous with the set of rational numbers.

There are infinite subsets of the set of irrational numbers that are equinumerous with the set of natural numbers.

There are infinite subsets of the set of irrational numbers that are equinumerous with the set of irrational numbers.

Any infinite subset of the set of natural numbers is equinumerous with the set of natural numbers.

If x is an infinite subset of the set of natural numbers and y is a finite subset of the set of natural numbers, then {n | n is in x and n is not in y} is equinumerous with the set of natural numbers.

If x is an infinite subset of the set of rational numbers and y is a finite subset of the set of rational numbers, then {r | r is in x and r is not in y} is equinumerous with the set of rational numbers.

If x is an infinite subset of the set of natural numbers and y is a finite subset of the set of natural numbers, then the union of x and y is equinumerous with the set of natural numbers.

If x is an infinite subset of the set of rational numbers and y is a finite subset of the set of rational numbers, then the union of x and y is equinumerous with the set of natural numbers.

'distance' is defined by the absolute value of the difference between points, not by cardinality. The distance between 0 and 1 is 1, no matter what about the cardinalities of the set of irrationals and the set of irrationals between 0 and 1.


That's all ordinary mathematics, proven from the ordinary axioms.

One is free to propose different axioms that prove differently.