One of the "proofs" of infinite series I remember being taught,

there is an infinite number of integer numbers,
there is an infinite number of fractional numbers between each integer,
hence there is two infinites, ( or more

Both statement are true, but both wrong,

the special numbers of infinity, and zero , do not behave as other numbers.


Jenifer ,
how do you square that circle ?

To square the circle, simply reduce those 'infinite' number of flat sides reconnecting the remaining ones from the internal gaps created and then grow each infinitesimal one by an infinite amount in such a way that they reach a unity unit , until only 4 equal length ones are left with opposing sides parallel to each other

 

I like your question on does the square root of 2 contain pi or does pi contain the square root of 2. I would think it would have to. But the larger quandary: if any irrational number contains an infinite amount of information, then can it contain itself? If it does, it has to be repeatable. If it does not where do we go with that. Seems like a logical contradiction.

There are some catches when asking does something contain itself. Apparently that will always cause a contradiction.
Lemma: F is a function that adds 1 to a function if the function does not add 1 to itself already.
So the function x adds 1 to itself to get x+1, and the function y+1 already adds 1 to itself so no need to add 1.
So what happens with the function F then? If F does not add 1 to itself, then it has to add 1 to itself, but if it adds 1 to itself then it cant add 1 to itself. Contradiction.
I hope i explained that right.

As to sqrt(2) containing itself, i wonder if we can say that a set A always contains itself because if it exists then it must contain all the members of itself. Of course then we might go on to stipulate that we are not allowed to use the entire set as containing itself we have to stipulate that it must be a subset of the original set that we bring into question, and so can we find any subset of sqrt(2) in the sqrt(2) and i think that has to be true,

Here is a rough idea of how the proof of the why there are more reals than natural numbers i hope i do this right.
Say we start with this set of random numbers that correspond to exactly 1 natural number (1 to 3):
[1] 0.154
[2] 0.783
[3] 0.834
Now if we take the n^th digit of each number (n increments as we go down the list) we get a new number. So first we take 0.1 from the first, then 0.08 from the second, then 0.004 from the third (this is really the digits 1, 8 and 4) and then form a new number by adding 1 to each digit, we get:
0.295
and this number must be different than any of those listed so there must be more reals than naturals.

Interesting

 

There are some catches when asking does something contain itself. Apparently that will always cause a contradiction.
Lemma: F is a function that adds 1 to a function if the function does not add 1 to itself already.
So the function x adds 1 to itself to get x+1, and the function y+1 already adds 1 to itself so no need to add 1.
So what happens with the function F then? If F does not add 1 to itself, then it has to add 1 to itself, but if it adds 1 to itself then it cant add 1 to itself. Contradiction.
I hope i explained that right.

As to sqrt(2) containing itself, i wonder if we can say that a set A always contains itself because if it exists then it must contain all the members of itself. Of course then we might go on to stipulate that we are not allowed to use the entire set as containing itself we have to stipulate that it must be a subset of the original set that we bring into question, and so can we find any subset of sqrt(2) in the sqrt(2) and i think that has to be true,

Here is a rough idea of how the proof of the why there are more reals than natural numbers i hope i do this right.
Say we start with this set of random numbers that correspond to exactly 1 natural number (1 to 3):
[1] 0.154
[2] 0.783
[3] 0.834
Now if we take the n^th digit of each number (n increments as we go down the list) we get a new number. So first we take 0.1 from the first, then 0.08 from the second, then 0.004 from the third (this is really the digits 1, 8 and 4) and then form a new number by adding 1 to each digit, we get:
0.295
and this number must be different than any of those listed so there must be more reals than naturals.

Interesting

That proof is very concrete. Thanks for sharing.

 

The L shaped matrix has practical functionality in order to make decisions, solve problems, and improve processes.
The nature of this data structure using diagrams can simplify casual relationships making it easier to expand into user defined relationships. When comparing to arithmetic functions it is the weight that each element is assigned that accumulates into an expression which can be fuzzy for some period.

For example one can select shoes to purchase after moving enough hangers back and forth and seeing the price tag but would'nt all those blouses match well with a certain pair of shoes ? which relates to cyber clothes shopping, shop until you drop episode resultng in a dispropotionate number of shoes in the closet vs blouses.

Computer science online can gather cookies which select data sets based on psychology, buying habits and emotions to name a few relationships
but they all feed into a function when comparing whole process it is a data selection subset it was a person that clicked on those links
and typed those search terms narrowing the process to the arithmetic transaction at the shopping cart . Thankyou for your purchase!

 

has any one heard from the OP in a while as to if we have answered their question ?

 

That proof is very concrete. Thanks for sharing.

Oh no problem. What else is interesting there is that there are an infinite number of natural numbers and an infinite set of reals, yet there are more reals than naturals. I think this may address the original poster's questions, but then it also addresses infinity in and of itself. Could infinity have some sort of implied second dimension. That's what i think we see here. The strange thing then is how we handle limits.

If the limit of a/(a+1) as a goes to infinity is 1, then we assume that infinity plus 1 is still equal to infinity, so how can one infinity be greater than the other. But there infinity seems to be a comparative tool rather than an absolute number of some kind, because the limit of a/(2*a+1) is 1/2, which implies that infinity multiplied by a constant is not always equal to infinity as was the case when a constant is added to infinity. I n fact, there two times infinity is two times larger than infinity, strange as that sounds.
Also the limit of a/(a^k+1) can be zero given k within some bounds, so a^k with k a positive constant is larger than just infinity. The limit of a/(a^a+1) as a goes to infinity is also interesting.

So with all that in mind, stating that infinity is the largest number possible may not be correct. The largest number possible may be impossible to define for each and every application we could ever encounter if we wish to try to define infinity in one way only so that it can be used in every single application we ever find without any modification.
Considering always the limit of a as ti goes to infinity, a is less than a*k, is leas than a^a, is less tahn a^a^a, is less than a^a^a^a, etc., which brings up the question what is the absolutely fastest changing (increasing) function we could ever find, which i think there is no answer because whenever we find a function that has a very large derivative we can always increase it by changing the function a little. So there can be no absolute infinity.

Ans so with all that in mind, it should be no wonder that infinity is not always the largest number we will ever encounter, and so having more reals than natural numbers even though they are both "infinite" in size should be at least possible because each set taken by themselves has an intrinsic characteristic of itself, but when compared to another set with a somewhat different intrinsic characteristic we find the relationship of those two characteristics to have a characteristic all it's own.
It's maybe like saying that we have a blue LED, and we have a red LED. No problem with that. But when we compare them, we get a result that we can express in different ways that do not have to be either red or blue but can be something else.

Havent thought about this stuff in a long time now but it's interesting to think about now and then.

 

Infinity is not a number, nor is it a "completed thing" like a "set": Infinity is the term we use to define "the capacity to generate a number." "Different size infinities" is therefore entirely incongruous and oxymoronic. The Axiom of Infinity, which says there is at least "one infinite set," and traced back to Georg Cantor's psych ward, is flawed because it's an oxymoron that uses "actual infinity" (insane rationalization of the irrational) instead of "potential infinity." All math was based on the latter before Georg Cantor and his Paradise Infirmary. ℝ is not a set: ℝ is a continuum, and ℕ is nothing but whole number cuts of it.

 

Just a note:

This has moved on a long way from the orriginal topic of
"
"Functions" in math vs. computer science
"

And is in the education forum

Yet we are now talking about thye concept of multiple infinities,

Seems to me that thios topic is closed, and a new one should be started, referring back to this one if so desired,
and should probably be in the maths forum.

 

Just a note:

This has moved on a long way from the orriginal topic of
"
"Functions" in math vs. computer science
"

And is in the education forum

Yet we are now talking about thye concept of multiple infinities,

Seems to me that thios topic is closed, and a new one should be started, referring back to this one if so desired,
and should probably be in the maths forum.

This is in the math forum.

Infinity’s relationship to functions in math vs. computer science is highly integral to the original post‘s contents. The very point of the post is how it relates to the notion of “infinite sets”, and to elicit comments about this very thing.

 

This is in the math forum.

Infinity’s relationship to functions in math vs. computer science is highly integral to the original post‘s contents. The very point of the post is how it relates to the notion of “infinite sets”, and to elicit comments about this very thing.

In that case

Can I ask,

how are you proposing to represent infinity in a computer as per your original title ?

Also, the end of your initial post has
" If we say |ℕ| has infinite cardinality and |ℝ| has infinite cardinality, the number of potential elements in both is equal, therefore |ℕ| = |ℝ|, and as can be seen in base infinity, |ℕ| is just as "large" as |ℝ|. "

which is what we all agree on,

number of integers is infinite,
number of fractions between any two integers is infinite

QED, infinity is a special concept for mathematical convenience, not something that can be put into a computer,



I would propose even more then that this post is off line with its initial posting,

and should be close, and a new post on infinity started in the maths forum.

 

In that case

Can I ask,

how are you proposing to represent infinity in a computer ?

I would propose even more then that this post is off line with its initial posting,

A thread can deviate a bit within the parent scope, which was made general enough here. Let the mods decide. They will let me know if it’s off-topic.

(your question involves computer science and infinity, so it’s still in the scope)

Just like there’s no technical representation of pi, or any other infinitesimal, irrational numeric expression (not technically a number), there is no actual representation of infinity in a computer (computing is done with numbers or bits, and numbers/bits are by definition finite). There is no “irrational” in a rational computer, there are only rational approximations. In which case, when we use “pi” in a computer, it’s technically not pi. It’s a rationalized, or quantized portion of the non-quantizable, sufficient for everyday calculation.

 

In that case

Can I ask,

how are you proposing to represent infinity in a computer as per your original title ?

Also, the end of your initial post has
" If we say |ℕ| has infinite cardinality and |ℝ| has infinite cardinality, the number of potential elements in both is equal, therefore |ℕ| = |ℝ|, and as can be seen in base infinity, |ℕ| is just as "large" as |ℝ|. "

which is what we all agree on,

No, the ZFC foundation of modern math would disagree, and claim multiple sizes of infinite sets called transfinites, where |ℕ| < |ℝ|. I’m contending this is spurious thinking when properly examined.

 

No, the ZFC foundation of modern math would disagree, and claim multiple sizes of infinite sets called transfinites, where |ℕ| < |ℝ|. I’m contending this is spurious thinking when properly examined.

Thank you

I assume your refering to this

https://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory

an you illuminate on the relationship between this and the functions in maths v computer science ?

 

Just a note:

This has moved on a long way from the orriginal topic of
"
"Functions" in math vs. computer science
"

And is in the education forum

Yet we are now talking about thye concept of multiple infinities,

Seems to me that thios topic is closed, and a new one should be started, referring back to this one if so desired,
and should probably be in the maths forum.

I have to disagree, and here is why.
If you have a premise A and its truth or falsity depends highly on premise B, then you almost HAVE to talk about B because there would be no way to prove if A is right or wrong without that.

So actually it is the converse of what you think, again. It should be talked about because if it is not then premise A may be presented as a false truth which readers would tend to believe.

This is a good example too because the original discussion uses properties of infinity as a basis for the entire argument. If you dont know what infinity is all about, you can never understand this thread properly.

 

Infinity is not a number, nor is it a "completed thing" like a "set": Infinity is the term we use to define "the capacity to generate a number." "Different size infinities" is therefore entirely incongruous and oxymoronic. The Axiom of Infinity, which says there is at least "one infinite set," and traced back to Georg Cantor's psych ward, is flawed because it's an oxymoron that uses "actual infinity" (insane rationalization of the irrational) instead of "potential infinity." All math was based on the latter before Georg Cantor and his Paradise Infirmary. ℝ is not a set: ℝ is a continuum, and ℕ is nothing but whole number cuts of it.

So what do you say about the proof that there are more real numbers than natural numbers?

I think you should also show some examples of what you talk about regardless what it is. There are too many things being discussed here that are incomplete without some solid evidence which can be simple examples. That way more people will understand where you are coming from.

 

Thank you

I assume your refering to this

https://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory

an you illuminate on the relationship between this and the functions in maths v computer science ?

Yes, which was based on Cantor’s theories. See the “diagonal argument.”

 

So what do you say about the proof that there are more real numbers than natural numbers?

I think you should also show some examples of what you talk about regardless what it is. There are too many things being discussed here that are incomplete without some solid evidence which can be simple examples. That way more people will understand where you are coming from.

It’s axiomatically based on a false notion that ℕ and ℝ are separate things, and that more than 1 continuum exists, when all of ℕ exists in ℝ. We can map all of ℝ to any 2 intervals of ℕ, like (0, 1), so ℕ contains the continuum and is just whole number cuts of it, like ”guard bands.” We can also model ℝ using 0 and 1. Therefore both hail from the same phenomena, and that is consciousness. They’re both names for potential infinity.

Ergo, consider a magic apple tree, that when you pick an apple and put it in a bushel, another apple appears in its place. You keep picking apples from the tree and putting it in the same bushel. ℝ is the tree, and ℕ is the bushel. Does it make sense to determine the cardinality of the tree by bijecting it to the bushel? No, they’re both the same thing in disguise, and treating them separately makes no sense.

 

I have to disagree, and here is why.
If you have a premise A and its truth or falsity depends highly on premise B, then you almost HAVE to talk about B because there would be no way to prove if A is right or wrong without that.

So actually it is the converse of what you think, again. It should be talked about because if it is not then premise A may be presented as a false truth which readers would tend to believe.

This is a good example too because the original discussion uses properties of infinity as a basis for the entire argument. If you dont know what infinity is all about, you can never understand this thread properly.

Hi @MrAi

There is truth in what you say, I agree that,

but

I'm looking from a users point, and trying to find an answer to a problem,

The original title says "functions in maths v computer science",

yet the question, though long, I believe boils down to

"If there is an infinite number of integers, and an infinite number of fractional between each integers, is one or other statement in error."

Hence my suggestion, that the post, is now off original topic.

and would be if greater to future users if a new post was started with a more descriptive title,

The alternative, is that OP are allowed to change the title as a post migrates to new areas.
which the OP has done before,
which makes all the original answers at best of no help, at worse, could mislead future readers.

 

Hi @MrAi

There is truth in what you say, I agree that,

but

I'm looking from a users point, and trying to find an answer to a problem,

The original title says "functions in maths v computer science",

yet the question, though long, I believe boils down to

"If there is an infinite number of integers, and an infinite number of fractional between each integers, is one or other statement in error."

Hence my suggestion, that the post, is now off original topic.

and would be if greater to future users if a new post was started with a more descriptive title,

The alternative, is that OP are allowed to change the title as a post migrates to new areas.
which the OP has done before,
which makes all the original answers at best of no help, at worse, could mislead future readers.

I’m not looking for “answers” here, just discussion on the matter. There was no specific question.

 

I’m not looking for “answers” here, just discussion on the matter. There was no specific question.

I have no problem with just wanting to sound out / discuss

But when I do, I try to make the heading clear on that subject

"Functions" in math vs. computer science

Seems to have no relationship to the way you seem to be taking this discussion,

hence my question,