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thinkandmull
Guest
So you think the “truth” of numbers as units is relative?
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thinkandmull
Guest
Whether using numbers or letters is irrelevant when considering that 1 3 5 do not line up with 1 2 3 4 5.
I don’t think there was anything strange in what I said about Cantor “larger infinities”; I am the one actually who is saying that that natural numbers are “larger” than the odd numbers.
I understand that infinity is not technically a number, but my position is that it is definite in sort of a way that a number is. There is a BASIC infinity. Take a larger and smaller infinity, anyone you like; then I ask you “by how much is the larger infinity greater”? However, does it really matter? Is it greater BECAUSE it has an infinity more? Why is not “one” or half of it sufficient?
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thinkandmull
Guest
There is still here the same flaw in logic. One-to-one correspondences **means **lining them up. If I put 1 to 1, 3 to 2, 5 to 3, 7 to 4, and 9 to 5, I’ve ignored 5 numbers in the larger series because, for example, I can put 1 3 and 5 along 1 2 and 3, that would be ignoring the existence of 2 and 4 then. Please answer this logic. The infinity part extended “out there” is not an equalizer on this.There is no “lining up”. We are discussing one-to-one correspondences. Bijections, if you prefer. Am I speaking a foreign language?
Yes or no: If f=2n-1, where n can assume integer values, is f a bijection? It must be, as it is a linear function with non-zero slope. Its range is the set of odd numbers, so there is a one-to-one correspondence between the integers and the odd numbers.
There is no “lining up”, so please stop trying to frame it that way.
Also, you never addressed my argument using the set {x, xyx, xyxyx,…}. How can both the sets of integers and the odds be the same size as this set but not be the same size as each other? Is your notion of cardinality not transitive?
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Oreoracle
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You’ve already been told several times that two sets, infinite or not, are regarded as being the same size if their elements can be placed in a one-to-one correspondence (a bijection). ** It is true that you have found a correspondence which is not one-to-one in your example of “lining them up”. But we require only a single bijection that works.** Not every relation need be a bijection for the sets to be of the same size.
But I think we’ve reached the real issue, which is that you think this notion of size and the notions of countable and uncountable infinities which follow have no merit. But we know that countable infinities behave in similar ways and that they are fundamentally different than uncountable infinities. If the real numbers formed a merely countable set, for example, then calculus would be impossible. That’s why we don’t just stick to the rational numbers. Indeed, uncountable infinities play an important role not just in analysis, but also in topology.
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thinkandmull
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Take one of **your ** larger and smaller infinities. Although you cannot “squeeze” the smaller infinity next to the larger by a 2n-1 function, however how can you show that the one is larger when for every number I take from the smaller infinity I can take one from the larger?
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thinkandmull
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This is just as with taken every odd or even number to “correspond” with every odd number doesn’t make the odd plus even equal to the odd
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rossum
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Mathematics is an axiomatic system. The truth or falsity of a statement depends on the set of axioms currently in play. Hence all mathematical statements are relative to the current sets of axioms.
For example, 1 + 1 = 2 is true in number base 3 and larger. In base 2 (i.e binary) the correct equivalent is 1 + 1 = 10.
Mathematical truths are relative the the current set of axioms. Change the axioms and some of the truths will change.
rossum
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rossum
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Of course not. A set with three elements cannot line up with a set of five elements.
However, that is not relevant to my question. I am not talking about 1, 3, 5 etc. I am talking about the set {nmd, sgqdd, ehud, rdudm, mhmd, dkdudm, sghqsddm, ehesddm, rdudmsddm, … }
Please answer the question I asked, not a different question.
rossum
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rossum
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You are assuming what you have to prove. You cannot assume that one series is “larger” without showing a proof.
We have shown proofs that the two sets are the same size because we can show a one-for-one match between the two sets.
Assuming what you need to prove is a logical error.
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thinkandmull
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“nmd, sgqdd, ehud, rdudm, mhmd, dkdudm, sghqsddm, ehesddm, rdudms” would be the most basic infinity. There are twice as much numbers in the odd plus even as there are even numbers.
Definitions can change, but my question was about the TRUTH of whether a negative times a negative equals a positive. If you are saying that this, and infinities, don’t essentially make sense on a certain deeper level, than I can accept your position, or really even keep my position, because what does it matter at that point? I can come to grips with that, as I have with Zeno’s paradox. What I’ve tried to do on this forum is make rational SENSE of the matter. However, some things in life may not make sense till we take our last breath. God, or whoever you believe in, bless.
P.S. Heisenberg in his book Physics and Philosophy said that in physics the idea of infinity leads to contradictions
Definitions can change, but my question was about the TRUTH of whether a negative times a negative equals a positive. If you are saying that this, and infinities, don’t essentially make sense on a certain deeper level, than I can accept your position, or really even keep my position, because what does it matter at that point? I can come to grips with that, as I have with Zeno’s paradox. What I’ve tried to do on this forum is make rational SENSE of the matter. However, some things in life may not make sense till we take our last breath. God, or whoever you believe in, bless.
P.S. Heisenberg in his book Physics and Philosophy said that in physics the idea of infinity leads to contradictions
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rossum
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Excellent, we are agreed. My set of letter sequences, {nmd, sgqdd, ehud, rdudm, mhmd, dkdudm, sghqsddm, ehesddm, rdudms …} can be put into a one-to-one correspondence with the natural numbers.
Now answer the second part of my question, how did I derive that sequence of letter-groups?
It does, but physics is not mathematics. Physics is not an axiomatic system; mathematics is. What applies to one may not apply to the other. This is the case with Heisenberg’s statement. It applies to physics but not to mathematics.
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thinkandmull
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If physics leads to contradiction’s with infinity, something is amiss in the experiments or the math. I was hoping that my arguments against Cantor would be the way to solve some of these difficulties, not that anyone would head to my arguments on this thread but maybe a mathematical error around for more than a century had been the root of confusion on this question even in physic application.
I am not sure if {nmd, sgqdd, ehud, rdudm, mhmd, dkdudm, sghqsddm, ehesddm, rdudms …} lines up with the odd numbers only, or every multiple of ten, or all the odd plus even. I admit I don’t know. How can one define the smallest of infinities, which I’m thinking your set is?
My mind has swirled around this evening on this question.
It led me to reconsider the case of two segments, one longer than the other. There is no one to one correspondence when they are parallel, but there is when the longer is made the hypotenuse of a triangle with the other one. This is assuming that there can be a one to one correspondence between things that don’t have a definite position since they are infinitely small. Still, the argument seems to hold
To read point A from B doing half-steps… well, A will never be reached, so the segment AB is not truly finite. But movement is real, so when can with one motion go from A to B. But how can an infinity of steps be made, and where is the last step?? Weird
I am not sure if {nmd, sgqdd, ehud, rdudm, mhmd, dkdudm, sghqsddm, ehesddm, rdudms …} lines up with the odd numbers only, or every multiple of ten, or all the odd plus even. I admit I don’t know. How can one define the smallest of infinities, which I’m thinking your set is?
My mind has swirled around this evening on this question.
It led me to reconsider the case of two segments, one longer than the other. There is no one to one correspondence when they are parallel, but there is when the longer is made the hypotenuse of a triangle with the other one. This is assuming that there can be a one to one correspondence between things that don’t have a definite position since they are infinitely small. Still, the argument seems to hold
To read point A from B doing half-steps… well, A will never be reached, so the segment AB is not truly finite. But movement is real, so when can with one motion go from A to B. But how can an infinity of steps be made, and where is the last step?? Weird
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hecd2
Guest
The set of {nmd, sgqdd, ehud, rdudm, mhmd, dkdudm, sghqsddm, ehesddm, rdudms …} is bntmszakx hmehmhsd
so it is bijective to the set of rationals, the set of natural numbers, the set of ODD numbers, and the site of all multiples of ten.
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thinkandmull
Guest
Ye it blurs the line between odd numbers and the odd plus even set. But its not logically acceptable for there to be equal infinite sets of both 1-10 and set 1 3 5 7 9, AND for there to be equal numbers in both series. The question may be asked why the set of odd plus even numbers is not “uncountable” then, since for every set compared, there are five more in the first series.
So apparently this is like Zeno’s paradox, and I shouldn’t have gotten into this discussion. Its a slipping slide into coocky town
So apparently this is like Zeno’s paradox, and I shouldn’t have gotten into this discussion. Its a slipping slide into coocky town
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yppop
Guest
Think…
Please explain how your inability to understand the concept of one-to-one correspondence (bijection) is like Zeno’s paradox. Which one?
Don’t worry about slipping into to coocky (you do mean kooky, don’t you) town; you’l feel right at home there!
Yppop
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thinkandmull
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Note to just copy and paste from my last two posts (
):
You have an equal amount of infinite sets, one containing 1 through 10 and the other set 1 3 5 7 9. You are saying that beyond there being equal numbers of sets, that the numbers in the sets are equal in the two series. However, what else defines an uncountable infinity than that for every one to one correspondence there is something that does correspond? For every single one to one correspondence of sets with these two series, there are 5 more numbers in a set in the first series.
This is why I said you have no basis for saying there is ever a “larger infinity”
Zeno’s paradox shows that trying to reach A from B by taking half-steps results in an infinity of such steps, and will never be completed., So the segment AB is not truly finite. Movement is real, there with one motion we can go from B to A. But how can an infinity of steps be made, and where is the last step here?? This is as logically strange as Cantors theory
):You have an equal amount of infinite sets, one containing 1 through 10 and the other set 1 3 5 7 9. You are saying that beyond there being equal numbers of sets, that the numbers in the sets are equal in the two series. However, what else defines an uncountable infinity than that for every one to one correspondence there is something that does correspond? For every single one to one correspondence of sets with these two series, there are 5 more numbers in a set in the first series.
This is why I said you have no basis for saying there is ever a “larger infinity”
Zeno’s paradox shows that trying to reach A from B by taking half-steps results in an infinity of such steps, and will never be completed., So the segment AB is not truly finite. Movement is real, there with one motion we can go from B to A. But how can an infinity of steps be made, and where is the last step here?? This is as logically strange as Cantors theory
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Tomdstone
Guest
Note to just copy and paste from my last two posts (
You have an equal amount of infinite sets, one containing 1 through 10 and the other set 1 3 5 7 9. You are saying that beyond there being equal numbers of sets, that the numbers in the sets are equal in the two series. However, what else defines an uncountable infinity than that for every one to one correspondence there is something that does correspond? For every single one to one correspondence of sets with these two series, there are 5 more numbers in a set in the first series.
This is why I said you have no basis for saying there is ever a “larger infinity”
Zeno’s paradox shows that trying to reach A from B by taking half-steps results in an infinity of such steps, and will never be completed., So the segment AB is not truly finite. Movement is real, there with one motion we can go from B to A. But how can an infinity of steps be made, and where is the last step here?? This is as logically strange as Cantors theory
- An uncountable set cannot be put in 1-1 correspondence with a countable set.
- The infinite series Sum(i=1 to infinity) 2^(-i) will sum to a finite number, namely 1.
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thinkandmull
Guest
You can 't put a one to one correspondence between the numbers in the sets 1 through 10, 11 through 20, ect. and the other sets 1 3 5 7 9, 11 13 15 17 19, ect. There is only a one to one correspondence between the sets, not the numbers in the sets. Why can’t you admit there is something true-correct about this argument? **The same principle **that makes one infinity larger than another is found here
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Tomdstone
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It has been explained n times why there is a 1-1 correspondence between {1,2,3,…} and {1,3,5,…}.
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thinkandmull
Guest
And I observed that what I just observed in my last post was STILL true. So we have another Zeno’s paradox
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