Infinity has gotten stranger

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Infinity has gotten stranger
I had posted that even though any two distinct transcendental numbers always has a rational number inbetween, the order of infinity of the real numbers (which contains the transcendental numbers) is one greater than that of the rational numbers.
Now hold onto your seats. The twin prime conjecture is no more. Yitang Zhang proved there are an infinite number of twin primes, even though they thin out as you go up the number line.
Infinity is a strange jungle (counterintuitive).
PhilX
Now hold onto your seats. The twin prime conjecture is no more. Yitang Zhang proved there are an infinite number of twin primes, even though they thin out as you go up the number line.
Infinity is a strange jungle (counterintuitive).
PhilX

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 Joined: Sun Aug 31, 2014 7:39 am
Re: Infinity has gotten stranger
I want to express an opinion as to why infinity is counterintuitive. It's because they're infinite sets, not numbers. There's nothing I know of that says the Aleph sets should behave like numbers. They operate differently from individual sets of numbers that are below Aleph zero.
My intuition tells me there may be more unusual properties with infinity. This is all I have to say on this for now.
PhilX
My intuition tells me there may be more unusual properties with infinity. This is all I have to say on this for now.
PhilX
 Lawrence Crocker
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Re: Infinity has gotten stranger
[quote][The twin prime conjecture is no more. Yitang Zhang proved there are an infinite number of twin primes, even though they thin out as you go up the number line. /quote]
This is not my understanding. I believe that extensions of Zhang's work have gotten us only to the point that there are infinitely many primes with gaps no greater than 248. Still a ways to go to show infinitely many of gap 2.
This is not my understanding. I believe that extensions of Zhang's work have gotten us only to the point that there are infinitely many primes with gaps no greater than 248. Still a ways to go to show infinitely many of gap 2.

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Re: Infinity has gotten stranger
I copied this from Wikipedia:
"On April 17, 2013, Yitang Zhang announced a proof that for some integer N that is less than 70 million, there are infinitely many pairs of primes that differ by N.[1][2] Zhang's paper was accepted by Annals of Mathematics in early May 2013."
PhilX
Note: some people may get technical and say there is a big difference between 70 million and two, but I feel that the spirit of Zhang's proof is what counts (btw the number has been reduced from 70 million to 246). The main point in my thread still stands, that infinity has properties when you look at it as a set rather than a number.
"On April 17, 2013, Yitang Zhang announced a proof that for some integer N that is less than 70 million, there are infinitely many pairs of primes that differ by N.[1][2] Zhang's paper was accepted by Annals of Mathematics in early May 2013."
PhilX
Note: some people may get technical and say there is a big difference between 70 million and two, but I feel that the spirit of Zhang's proof is what counts (btw the number has been reduced from 70 million to 246). The main point in my thread still stands, that infinity has properties when you look at it as a set rather than a number.
 Lawrence Crocker
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Re: Infinity has gotten stranger
Thanks for the correction. The current state of the art is 246, not 248. I do not want to deprecate Zhang's achievement. Still, there is a difference between 246 and 2. Looks as if there is hope for reducing it to gaps of 6, but the proof for the doubles may still elude for some time. I am not so clear on how this all bears on our understanding of the nature of infinity.
I am also not so clear on your set/number point. Are you only talking about cardinals? The transfinite ordinals have very different properties from the transfinite cardinals.
In your first post the proposition that the cardinality of the reals is "next" greatest after that of the rationals is the continuum hypothesis. Its denial is consistent with all standard set theories.
If you were interested in some musings on the philosophical implications of the apparent fact that the continuum hypothesis can neither be proved nor disproved, you might want to glance at: http://lawrencecrocker.blogspot.com/201 ... sand.html
I am also not so clear on your set/number point. Are you only talking about cardinals? The transfinite ordinals have very different properties from the transfinite cardinals.
In your first post the proposition that the cardinality of the reals is "next" greatest after that of the rationals is the continuum hypothesis. Its denial is consistent with all standard set theories.
If you were interested in some musings on the philosophical implications of the apparent fact that the continuum hypothesis can neither be proved nor disproved, you might want to glance at: http://lawrencecrocker.blogspot.com/201 ... sand.html
Re: Infinity has gotten stranger
While set theory may not equate to each set fundamentally being 1, a set can be observed as a "unit" implied as "1" in itself. So while a set may equal "x", it simultaneously must always equal "1" as (1,x).Philosophy Explorer wrote: ↑Thu Apr 23, 2015 7:18 pmI want to express an opinion as to why infinity is counterintuitive. It's because they're infinite sets, not numbers. There's nothing I know of that says the Aleph sets should behave like numbers. They operate differently from individual sets of numbers that are below Aleph zero.
My intuition tells me there may be more unusual properties with infinity. This is all I have to say on this for now.
PhilX
In these respects a set observes a dual role of unity and multiplicity.
If this is the case infinity is merely all number extending itself through 1, and 1 must maintain itself not only as the foundation of infinity but infinity itself as one does not only just extend through all number but contains all number. This is considering one takes the view of infinity as a set (or set of sets).
Re: Infinity has gotten stranger
Eodnhoj7 this is the first time you said something about math that I totally agree with.Eodnhoj7 wrote: ↑Thu Jan 11, 2018 11:28 pmWhile set theory may not equate to each set fundamentally being 1, a set can be observed as a "unit" implied as "1" in itself. So while a set may equal "x", it simultaneously must always equal "1" as (1,x).
In these respects a set observes a dual role of unity and multiplicity.
You are absolutely right. The essential aspect of a set is that it is a set. It is a single thing.
In the real world if we have two apples sitting on the teacher's desk; that's two apples.
A set is an abstraction in which we make the rather absurd and arbitrary claim that the two apples are in fact ONE. They are one set.
And the set itself is NOTHING ELSE BUT THOSE TWO APPLES. A set is entirely characterized by its elements.
Now in the real world, do we think there is a "set" of apples? I'm not so sure. I tend to doubt it.
A more striking example is when there is only one apple. In this case there is an apple; and there is a set containing that apple; and there is a set containing the set that contains that apple; and you do iterate this abstraction endlessly.
Do we really believe that the set containing the set containing the apple is a thing deserving of being said to exist? It's a damn good question IMO. I love set theory but I never confuse it with reality.
And finally, it gets worse. If we removed the apple, then the empty set is still there!
But yes you are absolutely correct. A set is a multiplicity AND a unity. That's exactly what it means to be a set. This is exactly right.
Re: Infinity has gotten stranger
wtf wrote: ↑Fri Jan 12, 2018 3:35 amEodnhoj7 this is the first time you said something about math that I totally agree with.Eodnhoj7 wrote: ↑Thu Jan 11, 2018 11:28 pmWhile set theory may not equate to each set fundamentally being 1, a set can be observed as a "unit" implied as "1" in itself. So while a set may equal "x", it simultaneously must always equal "1" as (1,x).
In these respects a set observes a dual role of unity and multiplicity.
?. The only disagreement I recall was the categorization of math itself, it is physical, abstract or both? It really depends if 1 is premised in space or is not. If one is premise in space as dimension through direction, with direction occurring relative to other directions, then 1 can be premised in the 1d line as "unit". We can observe this intuitively through the manifestation of the symbol of one being a line, with curvature or potential curvature, across many cultures.
In a separate respect we can observe this in the act of demarcation where we apply a line, as a dimension, to an object and in turn we manifest a unified "set" as the object itself, while simultaneously in a separate respect "separate" the unit object from other objects through this very same demarcation. In these respects the application of the 1d line observes unity and multiplicity as both "unit" and "units".
As the 1d line is abstract we can take measurement for what it is as strict produce of consciousness.
However in a separate respect, observe the foundation of the physical world as frequencies (particlewaves) or "strings" we can observe the line as relating to other lines through an angulature resulting in the 0d point. In these respects what we understand of the physical world may strictly be lines relating to other lines to form other lines.
However if the 1d lines is conducive to 1 as a spatial entity, it changes our understanding of "1" entirely as the 1d is axiomatic in both abstract and physical means. Space through dimension, is the root of all physical and abstract consciousness through consciousness relating to itself under a law of multiplicity/relation.
You are absolutely right. The essential aspect of a set is that it is a set. It is a single thing.
In the real world if we have two apples sitting on the teacher's desk; that's two apples.
A set is an abstraction in which we make the rather absurd and arbitrary claim that the two apples are in fact ONE. They are one set.
And the set itself is NOTHING ELSE BUT THOSE TWO APPLES. A set is entirely characterized by its elements.
Now in the real world, do we think there is a "set" of apples? I'm not so sure. I tend to doubt it.
A more striking example is when there is only one apple. In this case there is an apple; and there is a set containing that apple; and there is a set containing the set that contains that apple; and you do iterate this abstraction endlessly.
Do we really believe that the set containing the set containing the apple is a thing deserving of being said to exist? It's a damn good question IMO. I love set theory but I never confuse it with reality.
And finally, it gets worse. If we removed the apple, then the empty set is still there!
But yes you are absolutely correct. A set is a multiplicity AND a unity. That's exactly what it means to be a set. This is exactly right.
In these respects, what we understand of sets (although I do not believe set theory equates the set to "1" in itself) is in fact a duality conducive to two. In these respects, the question occurs relative to set theory, is the foundation of number rooted in 2? I do not agree that it is, however set theory appears to require this. If this is the case, set theory cannot provide the exact definition as to what number is as a polarity between unity and multiplicity occurs.
In a separate respect we can observe that "1" has a dual nature of "unity" and "unit". If one is to viewed qualitatively under the terms of "unity" we can observe that in many respects it must contain all numbers within it, with each number in itself being an extension of one. As one must contain all number, in these respects, it maintains a nature equal to infinity as "all number" is quantitatively numberless number (infinity is define in such terms). 1 in itself is both limit and nolimit in a separate respect.
If we view one as "unit" is must continually relate, as units to further units in order to exist.
I will expand later when I have the time.
Re: Infinity has gotten stranger
All I'm saying is that a set is regarded as a single entity; yet it is identical to a collection of many entities. In that sense a set is a duality between the one and the many.
Re: Infinity has gotten stranger
A hyuman is a set of individual cells.
A series is a collection of individual elements.
A human has qualities, physical and mental, emotional, different from the qualities of his or her cells.
A set has characteristics and traits, different from the elements it is made up of.

I don't like the expressions duality  singularity  plurality. The three serve no purpose, their categorizing capacity is extremely simplistic.
Re: Infinity has gotten stranger
I always thought so too; just something philosophy makes more complicated that in reality is barely a problem. Philosophy seems more often a clash between terms than actual ideas...except when the Greeks practiced it.
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