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for every number there is exactly one square; hence, there cannot be more of one than of the other. This is an early use, though not the first, of the idea of one-to-one correspondence in the context of infinite sets (wikipedia)

A = 1,2,3,4,5.....

B = 1,4,9,16,25

They say that A and B are both equally infinite which can not be true.

For A includes at least 9 infinities while B includes only one.

A include as part of that serie infinite set of each 9 numbers.

A includes

3333333333..... untill infinity while B does not.

66666666666..... infinite

777777777777...... infinite

And so on.

A is 9 x infinite at least

B is only 1 x infinite

Am I correct or is Galilei correct?

B can not include infinite amount number 3333333.... but A can.

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15 hours ago, Diamonds said:

for every number there is exactly one square; hence, there cannot be more of one than of the other. This is an early use, though not the first, of the idea of one-to-one correspondence in the context of infinite sets (wikipedia)

A = 1,2,3,4,5.....

B = 1,4,9,16,25

They say that A and B are both equally infinite which can not be true.

For A includes at least 9 infinities while B includes only one.

A include as part of that serie infinite set of each 9 numbers.

A includes

3333333333..... untill infinity while B does not.

66666666666..... infinite

777777777777...... infinite

And so on.

A is 9 x infinite at least

B is only 1 x infinite

Am I correct or is Galilei correct?

B can not include infinite amount number 3333333.... but A can.

Maybe I am wrong; but I don’t see what you presented as an accurate representation of Galileo's paradox:

Quoting Galileo:

“If I should ask further how many squares there are one might reply truly that there are as many as the corresponding number of roots, since every square has its own root and every root its own square, while no square has more than one root and no root more than one square [...]. But if I inquire how many roots there are, it cannot be denied that there are as many as the numbers because every number is the root of some square. This being granted, we must say that there are as many squares as there are numbers because they are just as numerous as their roots  and all the numbers are roots [...] Yet at the outset we said that there are many more numbers than squares, since the larger portion of them are not squares”

Note: There is no mention of including subsets such as 3333333333..... untill infinity

66666666666..... infinite

777777777777...... infinite

And so on, in the set of the natural numbers, n

Now take the infinite set of squares, n2: IBI={1,4,9,16,25,36,49,64,81...,n2,….}..

By inspection, it is possible to have a one-one mapping of the naturals to the squares of the naturals, meaning the two infinite sets have the same number of elements. IOW, they have equal numerosity.

This confirms the first part of the paradox is not problematic

Again, take the infinite set of naturals n:   IAI= {1,2,3,4,5,6,7,8,9….,n….}..

Now take the infinite set of naturals,n:  ICI= {1,..,..2,……….3,….,n….}..

By inspection there cannot be a one-one mapping between the infinite set of naturals and the infinite set of naturals. There would be empty spaces in ICI

Indicating there are more elements in IAI.

What I have attempted to show is there is no paradox. The way Galileo stated it: “there are many more numbers than squares, since the larger portion of them are not squares”

That last sentence I hope to show to be true but irrelevant to the first part, since there is a one-one mapping between the elements in IAI, n and elements in IBI, n2.

I don’t see a paradox since Galileo seems to comparing apples and oranges: On the one hand he is correct in saying there are as many squares as there are numbers because they are just as numerous as their roots.

However, when Galileo says: there are many more numbers than squares, since the larger portion of them are not squaresIn this case, he should be checking to see how many of n are squares. To do this he should be comparing set IAI with set ICI.

By inspection, there are more elements in IAI than in ICI, showing this is just a logical fallacy and not a paradox at all.

Of course, I expect someone who has just completed a course in set theory and linear algebra to totally destroy my simplistic explanation by posting equations with abstract symbols that I have forgotten the meaning of.

In fact, that is exactly what I am seeking: a nice fresher course in set theory, as long as it doesn’t end with me being sent to an insane asylum as happened to Georg Cantor.

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forum;

This is part of my current project to disprove the 'Cantor diagonal argument'.

The 1-1 correspondence is usually credited to Cantor, who uses it to prove there are 'infinite sets' with a greater cardinality (how many) than N the set of integers.

The problem is lack of consistency.

example 1: as many squares as integers,

before,

n {1,2,3, 4, 5, 6, 7, 8, 9...}

n2 {1,4,9,16,25,36,49,64,81...}

after removing redundant elements (comparing to themselves),

n {2,3,5,6,7,8,...}

n2 {...}

There is no 1-1 correspondence.

example 2: as many even integers as integers,

N {1,2,3,4, 5, 6...}

E {2,4,6,8,10,12...}

after removing redundant elements (comparing to themselves),

An example of a true 1-1 correspondence,

D {1,3,5,7, 9, 11...}

E {2,4,6,8,10,12...}

As many even integers as odd integers.

Verified by statistics, on average 1/2 of a random sampling are even.

Each set has unique elements.

Cantor's basic error is described here:

"… classical logic was abstracted from the mathematics of finite sets and their subsets …. Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. This is the Fall and original sin of [Cantor's] set theory …."
(Hermann Weyl, 1946)

Measurement requires boundaries. Infinite objects have none!

Cantor suffered from depression.

Here is latest paper 'Cantor Illusion.pdf'

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Three points to all here.......

1. You cannote expect to say anything intelligable about infinite sets  using small finite sets as an example;

2. Cantors paradox (so-called) refers only to the positve integers. These do not include 0.333..... and 0.9999.... for example

3. One-to-one correspondence cares not a jot for "gaps" or order. It merely means that for every X there exists a corresponding Y. The esiest way to see this is to index a "gappy" or unordered set e.g a(1), e(2),b(3) x(4) etc. There is no need (in fact it may be futile) to try and specify an inverse fuction

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On 1/27/2024 at 11:27 PM, QuarkH said:

Three points to all here.......

1. You cannote expect to say anything intelligable about infinite sets  using small finite sets as an example;

I’m not sure anyone said anything intelligible, LOL  But I am sure nobody was talking about small finite sets.

Those dots . . . . . .indicate the sets go on to infinity as in:

IAI={1,2,3,4,5,6,7,8,9….,n…..}..

IBI={1,4,9,16,25,36,49,64,81...,n2,….}..

ICI= {1,..,..2,……….3,….,n….}..

Quote

2. Cantors paradox (so-called) refers only to the positve integers. These do not include 0.333..... and 0.9999.... for example

I’m sure we are all well aware of that. Nobody here is using decimal fractions in this discussion.

Quote

3. One-to-one correspondence cares not a jot for "gaps" or order. It merely means that for every X there exists a corresponding Y. The esiest way to see this is to index a "gappy" or unordered set e.g a(1), e(2),b(3) x(4) etc. There is no need (in fact it may be futile) to try and specify an inverse fuction

I disagree. If there are gaps, there is no one-one correspondence.

For example:

Trying to get a one-to-one correspondence between  IAI and  ICI will create gaps at 3,5,6,7,8, which shows set IAI has more elements (more numerosity) than set  ICI.

IAI={1,2,3,4,5,6,7,8,9….,n…..}..

ICI= {1,..,..2,……….3,….,n….}..

If I am wrong, (very possible) please correct me with a copy & paste, including link.

Thank you for your input and welcome to our humble forum.

Edited by OceanBreeze
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QuarkH;

[quote]

2. Cantors paradox (so-called) refers only to the positve integers. These do not include 0.333..... and 0.9999.... for example

[/quote]

His theory concerns sets of numbers on the continuous 'real' number line.

If S is the set of squares then S is a subset of N.

S-S={}

N-S={2, 3, 5, ... t}=T where t is a non square.

They are not equal and there is no 1 to 1 match.

'Infinite' says nothing about cardinality or 'how many'.

It literally means without a limit/end/boundary, immeasurable in a math context.

It does not mean an astronomically large number that can't be recorded.

It's ultimately a contradiction of terms.

There is nothing in human experience to serve as a basis for its conception.

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• 2 weeks later...

There seems to be some confusion, and I confess I conributed to it. The original post was about Galileo's so-called paradox. Sluggo brought up Cantor who said something quite different. Let's see if I can help......

First this. Every set has least two subsets - the empty set and the whole set itself. This is unavoidable from the definition of a subset

Second, a set is said to be countable if it can be put in a one-to-one correspondence with a subset of the set of non-negative integers'

Since there are infinitely many non-negative integers, it follows that, not only can this set be desibribed as countably infinite, then so can any other countable set.

Cantor's so-called paradox stated in modern terms is that "a countably infinite set can be put in one-to-one correspondance with a proper subset of itself " (a proper subset is any subset that is not the whole set itself). Later mathematicians declared that a set is infinte if and only it can be put into on-to-one correspondance with a proper subset of itself. Note that is: "a "proper subset, not "any".

Cantor, on the other hand proved that there exist not only countable infinite sets but also uncountably infinite sets. His poof, often called his "diagnonal argument" is one of the most beautiful in all mathematics, and relies on the fact that elements a countable set can be listed in an order.

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A={1 2 3 4 5...}

B={1 4 9 16 25...}

A-B={2 3 5 6 7...}

B-B={}

A has more elements than B.

Comparison of two sets results in an outcome of equal or different.

The difference determines the greater cardinality, especially when there are identical elements in both sets.

This seems to favor Galileo's conclusion.

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I am sorry, but this makes no sense.

Suppose there exists a set C whose elements are squares of elements of B, so that C<B<A

Suppose further that there exists a set D whose elements are squares of elements of C, so that D<C<B<A.

Now continue, and ask how many iterations will it take to arrive at a set with 0 or perhaps just 1 elements.

Your "logic" says there must come such a point, which we know cannot be true.

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qh;

N={1 2 3 4 5 6 7 8 9 10 11 12 13...}

Which is more abundant, red or black?

Every integer can form an integer that is a square.

Some integers can form an integer that is a square root.

Edited by sluggo
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Sluggo, since you are either not reading or not understanding my posts, I don't feel like wasting any more of my time on this

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20 hours ago, QuarkH said:

Sluggo, since you are either not reading or not understanding my posts, I don't feel like wasting any more of my time on this

Hey Quark, in all fairness to Sluggo, and myself and anyone else who has trouble with this topic, please be more patient and explain your thinking, even if you need to "dumb it down" or takes us through it in baby steps.

I am not a mathematician, but am familiar with some fairly sophisticated math due to my work as an engineer.

Still, to most people, including myself, the concept of an infinite anything is tough to absorb, even though I do know how useful it is to have infinite as a limit in integral calculus and I also appreciate using infinite series to solve trigonometry, even though most people don't realize that is what their hand-held calculators are doing!

Then, when people talk of different size sets, but all of them are infinite, I feel like maybe a scotch and soda (make it a double) is called for.

What I am trying to say is, your contribution to this subject is appreciated, even if not everyone understands or agrees with you.

I make no pretensions of understanding all of it, but I did find it interesting and on topic.

Numerosities are something new, and I would appreciate having your input: do you find the concept useful?

Ocean

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qh;

Per definitions in set theory:

A=B only if AB and BA.

N={1 2 3 4 5...}

E={2 4 6 8 10...}

EN

N-E={1 3 5 7 9...}=not E=E'

E+E'=N

E<N

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• 2 weeks later...

Sluggo, you are confusing sets with their cardinality, which is just a number.

Look.....the ordinal numbers  describe the "position" of an element in an ordered set, regardless of the nature of these elements. Therefore they have to defined abstractly of set theoretic terms.

The empty set maybe written as { }, but I prefer the symbol Ø (you'll see why in a bit). This is the set with zero (0) elements. The set with just one (1) element is therefore written {Ø}. Paying lip service to ordinal arithmetic, we get to the next element (2) by "adding" them, which in set theory is called "set union"and is written as U, giving Ø U {Ø} as 2. And so on.

In general we say that  x U {x} is the successor set to the set x. In set theory, we deal in axioms. And the axiom of infinty suggests there exist sets which contain 0, and, whenever they contain x conatain all its successor sets.

The cardinal number associated to any set is in one-to one correspondence with the least (the intersection) of all its succcessor sets. This is of course the greatest finite number, if it exists

Therefore you need to prove that the successor set of all n is different from that of all n².

In simple language, that there exists either an n that is greater than any n², or vice versa.

Good luck with that!

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QH;

[quote]
Sluggo, you are confusing sets with their cardinality, which is just a number.
[/quote]

I'm familiar with set notation.

No, that was Cantor. He declared the cardinality of E (even numbers) is the same as N (integers) and the cardinality of D (odd numbers) is the same as N (integers).

Which contradicts this quote from his philosophical writing:

"Let M be the totality (n) of all finite numbers n, and M¢ the totality (2n) of all even numbers 2n. Here it is undeniably correct that M is richer in its entity, than M¢; M contains not only the even numbers, of which M¢  consists, but also the odd numbers M¢¢ . On the other hand it is just as unconditionally correct that the same cardinal number belongs to both the sets M and M¢. Both of these are certain, and neither stands in the way of the other if one heeds the distinction between reality and number."

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Sluggo, are you sure you fully understood the quote you posted? What he is saying is precisely what I said in my first post, namely that in a finite subset of the natural numbers, there are more elements that there are even numbers. He clearly believes that only finite sets are permitted in the "real world" (personally I think this is questionable), hence his first claim

He claims that on the second hand, in the abstract world of "numbers" i.e mathematics, where infinte sets are allowed, the cardinality of the natural numbers is equal to that of the even numbers.

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I don't accept Cantor's ideas of infinite sets. He declares infinite sets have an independent existence (without proof), and are complete things. By definition they are incomplete without a limit. His idea is a contradiction of terms.

His 1 to 1 correspondence is correct for some things, such as 'there are as many even integers as odd integers'.

D:{1 3 5 7 9 ...}

E:{2 4 6 8 10 ...}

There are no common elements.

We begin with finite sets from N, like {1 2 3 4}. We extend it by adding (n+1).

When does N become infinite?

Cantor claims his idea of 'transfinite' numbers was divine inspiration and he was a messenger. If true, when the paradoxes appeared, who was at fault? If Cantor, was God an accomplice? Could that have been a case for 'defamation of character'?

Statistics vs Cantor one to one.

Number of members within a constant interval of N, 1 thru 100.

Even, 50/100 = .50

Squares, 10/100 = .10

Cubes, 4/100 =.04

Cantor's 1 to 1 correspondence is not compatible with statistics.

It yields 1.00 for the above classes.

Statistics is applied math concepts to a broad range of human activities, used daily around the world.

After years of effort, I have a paper that exposes his errors in the diagonal argument.

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