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Is -(- infinity) the exact co-ordinate opposite of  - infinity ?  :beer-fresh:

 

 

Nice question! My answer is No, because  - (- infinity) is just another way of saying + infinity. Since both positive infinity and negative infinity are not numbers, (they are very useful concepts) they cannot be assigned exact coordinate positions on the extended real number line, so they cannot be considered to be exact coordinate opposites.

 

If they were exact coordinate opposites, then:  ∞ - ∞ would be zero, but in fact is left undefined.

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If they were exact coordinate opposites, then:  ∞ - ∞ would be zero, but in fact is left undefined.

Yes, and that is because infinities are not equal.  Consider the sets of numbers divisible by 1 and the sets of numbers divisible by 4.  Both sets are infinite, but the set of numbers divisible by one is larger than the set divisible by 4.

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Yes, and that is because infinities are not equal.  Consider the sets of numbers divisible by 1 and the sets of numbers divisible by 4.  Both sets are infinite, but the set of numbers divisible by one is larger than the set divisible by 4.

I think this is wrong. Presuming we are talking about Natural numbers, then all numbers divisible by 1 is the set of natural numbers, and since you can put those in a one-to-one correspondence with numbers divisible by 4, they have the same cardinality.

Cardinality

...

Definition 1: | A | = | B |

Two sets A and B have the same cardinality if there exists a bijection, that is, an injective and surjective function, from A to B. Such sets are said to be equipotent, equipollent, or equinumerous. This relationship can also be denoted A≈B or A~B.For example, the set E = {0, 2, 4, 6, ...} of non-negative even numbers has the same cardinality as the set N = {0, 1, 2, 3, ...} of natural numbers, since the function f(n) = 2n is a bijection from N to E. ...

Edited by Turtle
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I think this is wrong. Presuming we are talking about Natural numbers, then all numbers divisible by 1 is the set of natural numbers, and since you can put those in a one-to-one correspondence with numbers divisible by 4, they have the same cardinality.

Cardinality

Maybe I gave a poor example, but I was trying to quickly think of one set of numbers that would be infinite and still have to be larger than another infinite set.  If my example is wrong, I'm too lazy to come up with another .

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Maybe I gave a poor example, but I was trying to quickly think of one set of numbers that would be infinite and still have to be larger than another infinite set.  If my example is wrong, I'm too lazy to come up with another .

D'oh! Allow me. From the same article I linked:

Definition 2: | A | ≤ | B |

A has cardinality less than or equal to the cardinality of B if there exists an injective function from A into B.

Definition 3: | A | < | B |

A has cardinality strictly less than the cardinality of B if there is an injective function, but no bijective function, from A to B.For example, the set N of all natural numbers has cardinality strictly less than the cardinality of the set R of all real numbers , because the inclusion map i : N → R is injective, but it can be shown that there does not exist a bijective function from N to R (see Cantor's diagonal argument or Cantor's first uncountability proof).

If | A | ≤ | B | and | B | ≤ | A | then | A | = | B | (Cantor–Bernstein–Schroeder theorem). The axiom of choice is equivalent to the statement that | A | ≤ | B | or | B | ≤ | A | for every A,B. ...

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Maybe I gave a poor example, but I was trying to quickly think of one set of numbers that would be infinite and still have to be larger than another infinite set.  If my example is wrong, I'm too lazy to come up with another .

Watch that cantor doc. I think his problem was that he was trying to understand a lion, rather than understand a post-lion concept. It's like trying to understand Down in relation to a migrogravity environment: All the problems go away when you realise "the enemy gate is down".

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Farming guy, an wxample you were looking for is powersets of an infinite set. An example if the set is {1,2} then the powerset is the set of all subsets of the set: {{},{1},{2},{1,2}}.

So for an infinite set like Natural numbers, the powerset of the natural numbers has bigger cardinality than the set of Natural numbers, so if you want the infinity of elements in Natural numbers is smaller than the infinity of elements in the set of all subsets of Natural numbers.

See here for more:https://en.wikipedia.org/wiki/Power_set

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No thanks. Cantor went insane because of his considerations.

Did thinking about transfinite numbers cause Cantor to go insane, or did Cantor being insane cause him to think about transfinite numbers?

 

Seriously, though, I don’t think Cantor was insane, not in the same way that, say, Gödel was – that is to say, he wasn’t a paranoid schizophrenic. Rather, as about half the mathematical world was really out to get him, combined with the death of his youngest child at age 2, he just got discouraged and depressed, enough to be hospitalized a few times. He may have had bipolar disorder, though I think the evidence for that is scant. Gödel died of starvation because he believed his food was poisoned.

 

Like most Math students, I had a prolonged flirtation with Cantor and his [math]\omega[/math], [math]\aleph_0[/math], [math]\aleph_1[/math], [math]\aleph_{\alpha}[/math], [math]\aleph_{\omega}[/math] and [math]\Omega[/math] thing, a terrible and terribly mathematically pathological rabbit hole if ever there was one, eventually reaching a state of instinctive revulsion at it all, and becoming a happy convert to the cult of the finite. That Cantor seemed unable to write or talk at much length about [math]\Omega[/math] without equating it with God smacks of some sort of insanity, which I couldn’t satisfyingly wrap my mind around, to the point I felt myself actively trying not to. Plus, mixing Greek and Hebrew letters is just crazy, man!

 

I got disoriented and thought this thread was this one, so can’t use the mathic zinger I had prepared here. I’ll use it there instead. :)

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