Confusion with finite sets vs an infinite set of natural numbers

[joigus]

7 hours ago, Boltzmannbrain said:

Clue to what?

Exactly.

[Boltzmannbrain]

21 minutes ago, Genady said:

Correct. R(n) is finite. {R(n) | n∈N} is not.

Why does R(n) change to infinite when n is an element of N instead of just an n?

This is not a facetious question.  I think we have come to the absolute heart of my issue.  

49 minutes ago, pzkpfw said:

 

Just a few posts ago you wrote (my bold) "I forgot to put more ellipsis under the n for indefinite rows in the OP."

This is what I tried to show with:

1 { 1 }

2 { 1, 2 }

... { 1, 2, ... }

There is no single finite n in N that gives a set with no end. The list ( { 1, 2, ... } ) is infinite, and the row number is also infinite.

Try thinking of the list ( { 1, 2, ... } ) as X on a graph and the list of lists as the Y. It's unbounded on both axis.

I see what you are saying.  There are certainly many reasons to explain away my issue.  But there are still reasons that maintain my issue from being resolved.

[Genady]

1 minute ago, Boltzmannbrain said:

Why does R(n) change to infinite when n is an element of N instead of just an n?

R(n) does not change to infinite. 

R(n) and {R(n)} are different things.

The former contains numbers in the range [1, n]. The latter contains sets R(n) for all n's. 

The former is finite, the latter is not.

 

[wtf]

54 minutes ago, Boltzmannbrain said:

Why does R(n) change to infinite when n is an element of N instead of just an n?

This is not a facetious question.  I think we have come to the absolute heart of my issue.  

To the best of my understanding of this thread, the heart of your issue seems to be that you don't quite get the following idea:

Each of the individual natural numbers 0, 1, 2, 3, 4, 5, ... is itself a finite quantity.

And there are infinitely many of them. 

In other words there are infinitely many finite things. And for some reason you have trouble going back and forth between those two levels. The finitude of each of the natural numbers, and the endlessness, or infinitude, of the procession of all of them via the process of endlessly adding 1.

There are infinitely many natural numbers, and each of them are finite.

Likewise each R(n) = {1, 2, 3, ..., n} is a finite set; and there are infinitely many of the finite sets R(n), namely R(1), R(2), R(3), etc.

Hope this is helpful.

Edited May 3, 2023 by wtf

[Boltzmannbrain]

59 minutes ago, Genady said:

R(n) does not change to infinite. 

R(n) and {R(n)} are different things.

The former contains numbers in the range [1, n]. The latter contains sets R(n) for all n's. 

The former is finite, the latter is not.

 

 

Okay, thanks for your patience.  I forgot to take into account the set symbols.  +1

 

My same issue still lingers if you care to continue.  

 

You say, "R(n) is finite. {R(n) | n∈N} is not".  Is every R(n) finite in LIST?

[Genady]

Just now, Boltzmannbrain said:

Is every R(n) finite in LIST?

Yes.

[Boltzmannbrain]

27 minutes ago, wtf said:

To the best of my understanding of this thread, the heart of your issue seems to be that you don't quite get the following idea:

Each of the individual natural numbers 0, 1, 2, 3, 4, 5, ... is itself a finite quantity.

And there are infinitely many of them. 

In other words there are infinitely many finite things. And for some reason you have trouble going back and forth between those two levels. The finitude of each of the natural numbers, and the endlessness, or infinitude, of the procession of all of them via the process of endlessly adding 1.

There are infinitely many natural numbers, and each of them are finite.

Likewise each R(n) = {1, 2, 3, ..., n} is a finite set; and there are infinitely many of the finite sets R(n), namely R(1), R(2), R(3), etc.

Hope this is helpful.

@Genady just went over this with me.  What keeps me confused is how the symmetry below gets broken.  Just going from some set R to the set of natural numbers N does not make sense to me.  Let's continue to use each n∈N is mapped to set R(n)= {x∈N | 1≤x≤n} 

 

1 element in R --> n = 1

2 elements in R --> n = 2

3 elements in R --> n =3

4 elements in R --> n =4

.

.

.

infinite elements in N --> n = a finite number

 

 

Or put more generally,

finite --> finite

finite --> finite

finite --> finite

.

.

.

infinite --> finite

 

How this symmetry between the left side and the right side gets broken is my main issue.

[Genady]

11 minutes ago, Boltzmannbrain said:

infinite elements in N --> n = a finite number

12 minutes ago, Boltzmannbrain said:

infinite --> finite

This does not exist. It never happens. There is no such symmetry breaking.

There are only finite numbers of elements in R on the left side. As has been said above, R(n) is always finite.

There are only R's on the left, never N.

[Boltzmannbrain]

14 minutes ago, Genady said:

This does not exist. It never happens. There is no such symmetry breaking.

There are only finite numbers of elements in R on the left side. As has been said above, R(n) is always finite.

There are only R's on the left, never N.

I just wanted to show how the symmetry breaks when going from R (finite) to N (infinite)

Why can't I do this?

[Genady]

Just now, Boltzmannbrain said:

I just wanted to show how the symmetry breaks when going from R (finite) to N (infinite)

Why can't I do this?

Because it never goes from R to N.

[pzkpfw]

3 hours ago, Boltzmannbrain said:

@Genady just went over this with me.  What keeps me confused is how the symmetry below gets broken.  Just going from some set R to the set of natural numbers N does not make sense to me.  Let's continue to use each n∈N is mapped to set R(n)= {x∈N | 1≤x≤n} 

...

That starts with "n∈N". So n is a member of the set of Natural numbers. That set is infinite. Any given selected value for n will be finite, and can be plugged into the "1≤x≤n" to make a finite list of 1 to n; but that n can be selected from any of the Naturals ... and there are infinite of them.

[Boltzmannbrain]

13 hours ago, Genady said:

Because it never goes from R to N.

This may be the best way that I can state my confusion.

What I am trying to show is that the finite sets of natural numbers, that start at 1 and increase by 1, seem quite logical when comparing both sides (the number of elements versus the input n). 

For example 5 elements in the set implies an input of n = 5.  Infinite elements implies an input of n = ???

How do we make this asymmetrical jump to an infinite number of elements with finite inputs??? 

 

11 hours ago, pzkpfw said:

That starts with "n∈N". So n is a member of the set of Natural numbers. That set is infinite. Any given selected value for n will be finite, and can be plugged into the "1≤x≤n" to make a finite list of 1 to n; but that n can be selected from any of the Naturals ... and there are infinite of them.

I am not sure what you are saying that I don't agree with.

My issue is really quite simple.  

Take a set of natural numbers that start at 1 and increase by 1.  When this kind of set is finite, we know there must be a finite n that also equals the number of elements that it has.  When this kind of set has an infinite number of elements, we know there must be an infinite n that also equals the number of elements that it has.

That is where my logic is leading me (for better or worse).  It leads me to believe that there must be an infinitely large n in the set of all natural numbers. 

It is perfectly symmetrical and perfectly proportional.  Why should it even be wrong?

Edited May 3, 2023 by Boltzmannbrain

[Genady]

1 hour ago, Boltzmannbrain said:

Infinite elements implies an input of n = ???

There are never infinite elements and so no such n is needed.

 

1 hour ago, Boltzmannbrain said:

How do we make this asymmetrical jump to an infinite number of elements with finite inputs???

There is no such jump.

 

[Boltzmannbrain]

35 minutes ago, Genady said:

There are never infinite elements and so no such n is needed.

 

I was referring to a set equal to N.

 

Quote

There is no such jump.

 

I meant a jump in comparing the finite vs infinite sets that start at 1 and increase by 1. 

 

[Genady]

Just now, Boltzmannbrain said:

I was referring to a set equal to N.

I understand. There is no set equal to N in your construction.

 

1 minute ago, Boltzmannbrain said:

I meant a jump in comparing the finite vs infinite sets that start at 1 and increase by 1. 

There is nothing to compare to because there is no infinite set in your construction that starts at 1 and increases by 1.

[Boltzmannbrain]

8 minutes ago, Genady said:

I understand. There is no set equal to N in your construction.

 

There is nothing to compare to because there is no infinite set in your construction that starts at 1 and increases by 1.

A set that starts at 1, increases by 1 and has infinite elements has the properties of N.  Isn't that sufficient to construct N?

[Genady]

1 minute ago, Boltzmannbrain said:

A set that starts at 1, increases by 1 and has infinite elements has the properties of N.  Isn't that sufficient to construct N?

Yes, it is. But there is no such set in your construction. IOW, there is no set in your construction "that starts at 1, increases by 1 and has infinite elements."

[Boltzmannbrain]

9 minutes ago, Genady said:

Yes, it is. But there is no such set in your construction. IOW, there is no set in your construction "that starts at 1, increases by 1 and has infinite elements."

I think you are saying that we can't glean anything from the finite sets of the forementioned type (start at 1 and increase by 1), let's call type T, to the infinite sets of type T.  Is that accurate?  

Edited May 3, 2023 by Boltzmannbrain

[Genady]

1 hour ago, Boltzmannbrain said:

I think you are saying that we can't glean anything from the finite sets of the forementioned type (start at 1 and increase by 1), let's call type T, to the infinite sets of type T.  Is that accurate?  

If we define "a type T set" to be "a set of numbers from 1, increasing by 1, up to a finite number", then there is no infinite set of type T.

 

[Boltzmannbrain]

17 minutes ago, Genady said:

If we define "a type T set" to be "a set of numbers from 1, increasing by 1, up to a finite number", then there is no infinite set of type T.

 

I totally agree with what you are saying.  I am interested to know if you know exactly what it is about these last few posts that may seem unintuitive to me.  

[Genady]

Just now, Boltzmannbrain said:

I totally agree with what you are saying.  I am interested to know if you know exactly what it is about these last few posts that may seem unintuitive to me.  

No, I don't know.

[pzkpfw]

I'm certainly lost as to what the issue is. Removing the sets from it all (as I think the basic issue is perhaps more fundamental and they just add noise):

If we start writing "all" the natural numbers we start with 1, next is 2, then 3.

All nice finite numbers.

But there's no "last" natural number. We'll never write it down, even with infinite time, so we have notations, like maybe:

1, 2, 3, ...

So "1" and "2" and "3" are easy finite numbers. But "..." represents "infinity", it's not a specific value, here it's "all the values".

Yes, we go "finite, finite, finite, infinity" (using the terms from a post about a day ago), but how is this any kind of issue?

[studiot]

 

When you say 'increase by 1' what exactly do you mean ?

Are you counting or performing arithmetic ?

These two processes are very different things, obeying different rules.

 

In order to appreciate this here is a Wikipedia list of what you need to study, which demonstrates why I said it would take so long.

I note you didn't reply to that post.

List of statements independent of ZFC - Wikipedia

 

 

 

 

19 minutes ago, pzkpfw said:

I'm certainly lost as to what the issue is. Removing the sets from it all (as I think the basic issue is perhaps more fundamental and they just add noise):

If we start writing "all" the natural numbers we start with 1, next is 2, then 3.

All nice finite numbers.

But there's no "last" natural number. We'll never write it down, even with infinite time, so we have notations, like maybe:

1, 2, 3, ...

So "1" and "2" and "3" are easy finite numbers. But "..." represents "infinity", it's not a specific value, here it's "all the values".

Yes, we go "finite, finite, finite, infinity" (using the terms from a post about a day ago), but how is this any kind of issue?

 

It shouldn't be an issue once you realise this as things change once you include infinity.

It is an axiom of (finite) arithmetic that there is a unique number Y which has the property that when added to any number , X, the result is x

X + Y = X

This unique number is of course more familiarly known as zero.

However if you introduce infinity you also have

X + ∞ = X

Contradicting the axiom of 'additive identity' as this axiom is called.

 

+1 to wtf and pzkpfw

Edited May 3, 2023 by studiot

[Genady]

12 minutes ago, pzkpfw said:

I'm certainly lost as to what the issue is. Removing the sets from it all (as I think the basic issue is perhaps more fundamental and they just add noise):

If we start writing "all" the natural numbers we start with 1, next is 2, then 3.

All nice finite numbers.

But there's no "last" natural number. We'll never write it down, even with infinite time, so we have notations, like maybe:

1, 2, 3, ...

So "1" and "2" and "3" are easy finite numbers. But "..." represents "infinity", it's not a specific value, here it's "all the values".

Yes, we go "finite, finite, finite, infinity" (using the terms from a post about a day ago), but how is this any kind of issue?

I certainly agree with you that there is no issue, and I don't try to guess what the fundamental misunderstanding is, but the OP asked to point to an error in his construction, if there is one. I use sets to formalize his construction to point out the error.

[Boltzmannbrain]

44 minutes ago, pzkpfw said:

I'm certainly lost as to what the issue is. Removing the sets from it all (as I think the basic issue is perhaps more fundamental and they just add noise):

If we start writing "all" the natural numbers we start with 1, next is 2, then 3.

All nice finite numbers.

But there's no "last" natural number. We'll never write it down, even with infinite time, so we have notations, like maybe:

1, 2, 3, ...

So "1" and "2" and "3" are easy finite numbers. But "..." represents "infinity", it's not a specific value, here it's "all the values".

Yes, we go "finite, finite, finite, infinity" (using the terms from a post about a day ago), but how is this any kind of issue?

 

Good idea, I will remove the sets from it, especially since it doesn't seem to be getting me anywhere.

Imagine a meter of paint is painted for every natural number that is in the set of natural numbers N.  Every n is finite, so how can the paint be infinitely far away?

It isn't a proof, but do you at least see how there is something that deserves attention?

1 hour ago, studiot said:

 

When you say 'increase by 1' what exactly do you mean ?

Are you counting or performing arithmetic ?

These two processes are very different things, obeying different rules.

 

Interesting (+1)  I suppose either, but I am not sure what counting means in a formal sense.

 

Quote

 

In order to appreciate this here is a Wikipedia list of what you need to study, which demonstrates why I said it would take so long.

I note you didn't reply to that post.

List of statements independent of ZFC - Wikipedia

 

 

 

 

I think you might have missed it because I never got a response to my response.  It is the 13th post down on the 3rd page of this thread.

 

Quote

 

It shouldn't be an issue once you realise this as things change once you include infinity.

It is an axiom of (finite) arithmetic that there is a unique number Y which has the property that when added to any number , X, the result is x

X + Y = X

This unique number is of course more familiarly known as zero.

However if you introduce infinity you also have

X + ∞ = X

Contradicting the axiom of 'additive identity' as this axiom is called.

 

 

Okay, I did ask you a question on that lost post.  It is somewhat relevant to what you are saying here.