God And Infinity, Worlds Without Number Are Numbered To God?

[Rob Osborn]

You claim it is on the list when you apply the "correct" rules, but you can't identify where.

But neither can Cantor. This of course is the whole issue. Knowing through logic it is in the list is all that matters. We both know that there is no real number that is infinity so it must exist at a limit we design otherwise it can't be known other than in principle.

[Analytics]

As far as I know, a real number is best represnted as any point in a measuring system whereas a natural number is a number used for counting something like when I count sheep at night.

IF I built a list where each new row was a different number how many rows could I build if it was possible to do it all at once. I know this is impossible but how many?

Mathematicians would say that real numbers don't exist on measuring systems; they say real numbers are theoretical points on a theoretical line. This might be the major difference in how you think and how mathematicians think. They are thinking about theretical points and lines, while you are thinking about measuring systems in the physical world.

Addressing your question, "how many rows" doesn't have an answer. When we say there are an infinite number of rows, we aren't saying "infinity" is "how many" rows exist, we are saying that the rows can't be counted because they don't end. It's fine to snap your fingers and say that they are there without end, but "how many" doesn't have an answer because you can't count them.

But neither can Cantor. This of course is the whole issue. Knowing through logic it is in the list is all that matters. We both know that there is no real number that is infinity so it must exist at a limit we design otherwise it can't be known other than in principle.

We don't know through logic that it is on the list. We know through logic that it isn't.

Just for fun, let's try again.

Assume you snap your fingers and create an endless list that contains all real numbers.

In response, I snap my fingers and create a single real number with an endless number of decimal places. Is my number on your list? By construction, it can't be the first number on your list, because the first decimal point has a different value. It can't be the second number, because the second decimal point has a different value, it can't be the third, because the third decimal point has a different value.

Look at any one of the uncountable items on your list, and I can prove that that item is different than my number, because the corresponding decimal places don't have the same value.

My real number isn't on your list, therefore the assumption that your list has all real numbers is false.

Edited February 1, 2012 by Analytics

[Zeta-Flux]

I can do even better than just create a list of all rationals. I can create a list of rationals, + the square roots and cube roots of rationals as well as pi and e. I'll even include rational multiples of these. In fact the list will contain sums, differences, and powers of rationals and many irrationals and much more (say, several complex numbers and even quaternions). How can I do it? I'll just use the Goedel number of each one. A Goedel number exists for pretty much any irrational number we ever talk about, but there are far more irrational numbers than Goedel numbers.

asbestosman,

Just to clarify, the notion of a Godel number is a metamathematical notion. You certainly can number things that way, but only in a meta-theory. In other words, a theory strictly stronger than the one you were considering in the first place.

The thing about Cantor's argument is that it works in the base theory. It shows that there simply is no bijection from the natural numbers to the real numbers, inside the theory. However, outside the theory one can in fact model the reals with a countable set. (It just is the case that the bijection showing its countability is not defined in the base theory.)

So you have to be very careful appealing the Godel numberings. However, in essence, what you wrote holds true.

[Rob Osborn]

Mathematicians would say that real numbers don't exist on measuring systems; they say real numbers are theoretical points on a theoretical line. This might be the major difference in how you think and how mathematicians think. They are thinking about theretical points and lines, while you are thinking about measuring systems in the physical world.

Well, regardless how you or I say it or define it, its all semantics. We both know what a real number is.

Addressing your question, "how many rows" doesn't have an answer. When we say there are an infinite number of rows, we aren't saying "infinity" is "how many" rows exist, we are saying that the rows can't be counted because they don't end. It's fine to snap your fingers and say that they are there without end, but "how many" doesn't have an answer because you can't count them.

So, if you can't count them, then the ".1111..." in my list is a just representation of what happens in the list when you go forever and can;t count them. Thanks for answering that one. It just happens to be the same number that Cantor is making. I rest my argument with that.

We don't know through logic that it is on the list. We know through logic that it isn't.

Just for fun, let's try again.

Assume you snap your fingers and create an endless list that contains all real numbers.

In response, I snap my fingers and create a single real number with an endless number of decimal places. Is my number on your list? By construction, it can't be the first number on your list, because the first decimal point has a different value. It can't be the second number, because the second decimal point has a different value, it can't be the third, because the third decimal point has a different value.

Look at any one of the uncountable items on your list, and I can prove that that item is different than my number, because the corresponding decimal places don't have the same value.

My real number isn't on your list, therefore the assumption that your list has all real numbers is false.

Ok, prove it. This whole argument is just going in circles. There is no math proofs to show that you can make a number not already in my list. Why do i say this? Because a number is only a number if it is finished! The Cantor number is never finished, it just goes on forever without end. That, by definition, is not a "number".

[Rob Osborn]

Here is a proof for you- The notation of making an abstract "mark" on paper can be seen as a natural number regardless of what it means, that is, if we are counting abstract marks, whatever they may be. So, every mark in any list of abstract numbers, including any diagonally formed new notation or Cantor number, automatically creates the one-to-one correspondence.

[asbestosman]

Zeta-Flux,

What if instead of Godel numbers, I just use the ASCII codes of the number (words like limit, or even English words describing the number apply) and convert those to an integer by lining them up next to each other and interpreting the binary / hex value of it?

Does that count as a meta-theory since it now involves an element of interpretation of symbols, or can it still work within the theory since it creates a surjection--in this case from many reals to English names / descriptions of the number, to ASCII, to a binary number?

By the way, I too have heard about a possibly way to make the reals well-ordered (basically all put in a list without missing any) outside of standard ZFC but I have never seen any plausible way of doing so. I don't think even Godel numbers do that--at least not that I know of. I'm certainly no expert in meta-mathematical considerations though. The thing that has me stumped is what Godel's Bew. (Provable) function would even look like in general. All I ever read is that it's a complicated expression but it definitely exists as an expression and is a finite Godel number.

Edited February 2, 2012 by asbestosman

[asbestosman]

Ok, prove it. This whole argument is just going in circles. There is no math proofs to show that you can make a number not already in my list. Why do i say this? Because a number is only a number if it is finished! The Cantor number is never finished, it just goes on forever without end. That, by definition, is not a "number".

What do you mean by "finished?": Is pi a number? The square root of 2? The Cantor number is certainly a number. It is defined as the limit result of something which converges. The square root of 2 can be defined the same way as can pi. These are all numbers.

[asbestosman]

Here is a proof for you- The notation of making an abstract "mark" on paper can be seen as a natural number regardless of what it means, that is, if we are counting abstract marks, whatever they may be. So, every mark in any list of abstract numbers, including any diagonally formed new notation or Cantor number, automatically creates the one-to-one correspondence.

No. The Cantor number is defined as the number derived from the limit of an algorithm. Your list may be constructed as an algorithm, but the limiting case does not apply to it since it contains no greatest element.

There is a difference in math between a closed circle and an open circle. An open circle does not contain its boundary. An open circle has the special property that every point within the circle has a non-zero radius which lies entirely within the circle. A closed circle contains its boundary. The boundary consists of those points for which a non-zero radius always contains points not within the circle. Google it for some pics.

The point about this open / closed sets (boundaries or no boundaries) is to help you visualize your list. Your list may well converge to 0.111111..., but it never actually reaches that point. It is open on that end.

[asbestosman]

Analytics,

Minor quibble from me. When I talk of lists, I only think of them in terms of a countable number of elements. If a set is uncountable, then I only speak of it as a set. Thus as far as I'm concerned, the largest a list can be is countable infinity or aleph_0. A list to me implies separate rows each of which could be assigned an integer. Sets, on the other hand, need not have any notion of a next element in the set.

Edited to add: Actually the reals can be put in a Well-order, so my quibble was wrong. The notion of a list of sorts (well-ordering) can apply to the reals in ZFC.

Edited February 3, 2012 by asbestosman

[Analytics]

Well, regardless how you or I say it or define it, its all semantics. We both know what a real number is.

I'm not sure we do. You talk about real numbers being "created" and being "finished" or "unfinished". People who know what real numbers are know that the concept of an "unfinished" real number doesn't make sense.

So, if you can't count them, then the ".1111..." in my list is a just representation of what happens in the list when you go forever and can;t count them. Thanks for answering that one. It just happens to be the same number that Cantor is making. I rest my argument with that.

You are conflating two things here. The list itself has no end. But all of the elements on the list have ends.

Ok, prove it. This whole argument is just going in circles. There is no math proofs to show that you can make a number not already in my list. Why do i say this? Because a number is only a number if it is finished! The Cantor number is never finished, it just goes on forever without end. That, by definition, is not a "number".

I'm not "making" a number which is "unfinished" until I'm done making it. I'm identifying a number that already exists.

Here is a proof for you- The notation of making an abstract "mark" on paper can be seen as a natural number regardless of what it means, that is, if we are counting abstract marks, whatever they may be. So, every mark in any list of abstract numbers, including any diagonally formed new notation or Cantor number, automatically creates the one-to-one correspondence.

That's a proof of what, exactly?

Analytics,

Minor quibble from me. When I talk of lists, I only think of them in terms of a countable number of elements. If a set is uncountable, then I only speak of it as a set. Thus as far as I'm concerned, the largest a list can be is countable infinity or aleph_0. A list to me implies separate rows each of which could be assigned an integer. Sets, on the other hand, need not have any notion of a next element in the set.

Thanks for the feedback abestosman. To be sure I understand this distinction, the set of all rational numbers can be put into a "list", but the the set of all real numbers cannot, correct?

Edited February 2, 2012 by Analytics

[Rob Osborn]

No. The Cantor number is defined as the number derived from the limit of an algorithm. Your list may be constructed as an algorithm, but the limiting case does not apply to it since it contains no greatest element.

There is a difference in math between a closed circle and an open circle. An open circle does not contain its boundary. An open circle has the special property that every point within the circle has a non-zero radius which lies entirely within the circle. A closed circle contains its boundary. The boundary consists of those points for which a non-zero radius always contains points not within the circle. Google it for some pics.

The point about this open / closed sets (boundaries or no boundaries) is to help you visualize your list. Your list may well converge to 0.111111..., but it never actually reaches that point. It is open on that end.

The new Cantor number and the number in my list are the same thing, that is my point.

[Rob Osborn]

What do you mean by "finished?": Is pi a number? The square root of 2? The Cantor number is certainly a number. It is defined as the limit result of something which converges. The square root of 2 can be defined the same way as can pi. These are all numbers.

Ok then, where is the new number finished? How many elements are in it in comparison to the list?

[Analytics]

The new Cantor number and the number in my list are the same thing, that is my point.

The number on your list is in position q on your list. The qth decimal place of that number has a value different than the value of the qth decimal place of Cantor’s number. Thus they aren’t the same number. That is true for every q.

How can you not see this? If the two numbers have different values in even a single decimal place, how could it possibly be the same number?

[Analytics]

Ok then, where is the new number finished? How many elements are in it in comparison to the list?

Again, what are you talking about when you say a number is "finished"? Like all numbers represented in a decimal system, there are an infinite number of decimal places to Cantor's number.

[Rob Osborn]

Again, what are you talking about when you say a number is "finished"? Like all numbers represented in a decimal system, there are an infinite number of decimal places to Cantor's number.

Then there must be an infinite number of "1's"/decimal places in the number ".1111..." in my list

[Analytics]

Then there must be an infinite number of "1's"/decimal places in the number ".1111..." in my list

Nope. Every number on your list has a finite number of 1's followed by an infinite number of 0's.

[Tarski]

Analytics, on 01 February 2012 - 09:25 AM, said:

You claim it is on the list when you apply the "correct" rules, but you can't identify where.

But neither can Cantor.

What?

Why should Cantor be able to tell where on the list his number is. His claim is that it is NOT on the list.

You still don't follow his argument obviously.

We both know that there is no real number that is infinity so it must exist at a limit we design otherwise it can't be known other than in principle.

The word "real" in the phrase "real number" does not refer to whether the number is real in the sense of existing in reality or not. It is a technical term. The real numbers are devised in order to do various things such as include pi. The reals are a sort of completion of the rationals. Ths is all stuff you don't know about.

Now, you said that infinity is not a real number. Well, no one every said it was a "real number" in the above sense.

We said that the cardinality of the set of real numbers (completion of the rationals) is larger than the cardinality of the rationals. The notion of cardinality extends our ordinary notion of the size of a set so as to make sense for infinite sets but does so in a very specific way.

If you can't say more precisely what cardinality refers to other than to say "size" or "number" then you don't get it yet.

Hint: It is about the existence or nonexistence of bijections.

[Zeta-Flux]

Zeta-Flux,

What if instead of Godel numbers, I just use the ASCII codes of the number (words like limit, or even English words describing the number apply) and convert those to an integer by lining them up next to each other and interpreting the binary / hex value of it?

Does that count as a meta-theory since it now involves an element of interpretation of symbols, or can it still work within the theory since it creates a surjection--in this case from many reals to English names / descriptions of the number, to ASCII, to a binary number?

By the way, I too have heard about a possibly way to make the reals well-ordered (basically all put in a list without missing any) outside of standard ZFC but I have never seen any plausible way of doing so. I don't think even Godel numbers do that--at least not that I know of. I'm certainly no expert in meta-mathematical considerations though. The thing that has me stumped is what Godel's Bew. (Provable) function would even look like in general. All I ever read is that it's a complicated expression but it definitely exists as an expression and is a finite Godel number.

I think you run into trouble trying to map English descriptions because there are certain descriptions which you cannot prove they really describe real numbers, nor can you disprove they describe real numbers. I suppose you could limit yourself to numbers which you can prove quickly do describe real numbers. But, frankly, it is simply easier to just list the numbers you care about and then prove directly they are countable (or use general principles to prove that you can do so).

The Lowenheim-Skolem theorems are not too difficult to understand, from what I've been told. You would just need to take a year or two of logic. I myself have not done so, and thus I also do not fully understand how they work either.

[Tarski]

The Lowenheim-Skolem theorems are not too difficult to understand, from what I've been told. You would just need to take a year or two of logic. I myself have not done so, and thus I also do not fully understand how they work either.

I thought you were a model theorist.

Anyway you might enjoy this:

http://www.nd.edu/~t...pers/spmath.pdf

and part 3 of this:

http://plato.stanford.edu/entries/paradox-skolem/

Apparently, there are some philosophical worries.

Edited February 3, 2012 by Tarski

[Zeta-Flux]

Tarski,

Nope, I'm a ring theorist (and occasional number theorist). Thank you for those articles. They were quite enlightening.

Cheers,

Zeta-Flux

[Rob Osborn]

What?

Why should Cantor be able to tell where on the list his number is. His claim is that it is NOT on the list.

You still don't follow his argument obviously.

The word "real" in the phrase "real number" does not refer to whether the number is real in the sense of existing in reality or not. It is a technical term. The real numbers are devised in order to do various things such as include pi. The reals are a sort of completion of the rationals. Ths is all stuff you don't know about.

Now, you said that infinity is not a real number. Well, no one every said it was a "real number" in the above sense.

We said that the cardinality of the set of real numbers (completion of the rationals) is larger than the cardinality of the rationals. The notion of cardinality extends our ordinary notion of the size of a set so as to make sense for infinite sets but does so in a very specific way.

If you can't say more precisely what cardinality refers to other than to say "size" or "number" then you don't get it yet.

Hint: It is about the existence or nonexistence of bijections.

This is all BS! These abstracts exist nowhere in reality. Its all in meaningless imaginations. In reality, there is no such thing as proving different sizes of infinity. I know it is a bunk idea and anyone with common sense will know it too. If two sets are being measured and one is bigger than another, then the smaller one cannot be infinite in size and must have an "end" or "run out". We know that infinity has no potential to either end or run out though so we know that the statement of two different sizes of infinity is a contradiction and thus false.

BTW Tarski, you could get a lot further in life if you were a little less condescending.

[asbestosman]

Thanks for the feedback abestosman.  To be sure I understand this distinction, the set of all rational numbers can be put into a "list", but the the set of all real numbers cannot, correct?

Edited: Actually I was wrong. Reals can be put in a well-ordering in ZFC, but I know of no formula for doing so.

One strange things with well-orders is that you can actually have more than one element of the order that has no unique predecessor (previous row) but all will have a unique successor (next row). I've been looking this stuff up and there's a concept called Ordinal-indexed sequences. However, there is also a concept or ordinary sequences which was more in line with what I initially had in mind with lists.

Even guys who are usually good at math can get things wrong. This is why we need technical terms so we can use precise language to convey meaning and avoid pitfalls of unguided intuition. See that Rob, even I make math mistakes.

Edited February 3, 2012 by asbestosman

[Analytics]

This is all BS! These abstracts exist nowhere in reality. Its all in meaningless imaginations. In reality, there is no such thing as proving different sizes of infinity. I know it is a bunk idea and anyone with common sense will know it too....

And yet the mathematical proof remains.

[Tarski]

This is all BS! These abstracts exist nowhere in reality.

does pi exist in reality?

Its all in meaningless imaginations.

and yet on we go with our real number system doing calculus and quantum physics getting things done.

I know it is a bunk idea and anyone with common sense will know it too.

common sense must uncommon since I personally know of no one that resists it once it is explained. (except you)

If two sets are being measured and one is bigger than another, then the smaller one cannot be infinite in size and must have an "end" or "run out".

We know that infinity has no potential to either end or run out though so we know that the statement of two different sizes of infinity is a contradiction and thus false.

After all this you are going back to this misguided intuition?

BTW Tarski, you could get a lot further in life if you were a little less condescending.

How far did I get?

Edited February 3, 2012 by Tarski

[asbestosman]

How far did I get?

Apparently neither to infinity nor beyond.

As a side note: I've edited a couple of my previous posts. I wasn't previously quite as clear in my thoughts on lists between sequences and transfinite sequences.

Edited February 3, 2012 by asbestosman