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Arthur T Knackerbracket has found the following story:
With a surprising new proof, two young mathematicians have found a bridge across the finite-infinite divide, helping at the same time to map this strange boundary.
The boundary does not pass between some huge finite number and the next, infinitely large one. Rather, it separates two kinds of mathematical statements: "finitistic" ones, which can be proved without invoking the concept of infinity, and "infinitistic" ones, which rest on the assumption — not evident in nature — that infinite objects exist.
Mapping and understanding this division is "at the heart of mathematical logic," said Theodore Slaman, a professor of mathematics at the University of California, Berkeley. This endeavor leads directly to questions of mathematical objectivity, the meaning of infinity and the relationship between mathematics and physical reality.
More concretely, the new proof settles a question that has eluded top experts for two decades: the classification of a statement known as "Ramsey's theorem for pairs," or RT22. Whereas almost all theorems can be shown to be equivalent to one of a handful of major systems of logic — sets of starting assumptions that may or may not include infinity, and which span the finite-infinite divide — RT22 falls between these lines. "This is an extremely exceptional case," said Ulrich Kohlenbach, a professor of mathematics at the Technical University of Darmstadt in Germany. "That's why it's so interesting."
The abstract is available on arXiv — the full article is available as a pdf.
[Ed note: Not a new story but interesting and will hopefully spark some discussion.]
-- submitted from IRC
(Score: 2) by Wootery on Tuesday August 01 2017, @08:20AM (8 children)
Neither makes any sense, because division by zero isn't defined.
One might be tempted to think that 0/0 = (1/0) * 0 = 0, but that doesn't work. [brilliant.org]
You can't say that, because division by zero is undefined. It doesn't matter if it strikes you as intuitive.
That doesn't work either. See my link above.
No, that question doesn't arise, because division by zero is undefined, and square root is only defined for finite values.
Yes, see Cantor's Diagonal Argument, but that's not relevant here. Again, division of one (finite) real number by any (finite) nonzero real number, always produces a finite result. It isn't meaningful to hypothesise about what division by zero 'would' produce.
I'm not an expert on infinities, but I believe they can never be given finite ratios. The cardinality of even numbers is the same as the cardinality of natural numbers, for instance. (And that's a different kind of infinity [wikipedia.org] than would be relevant here anyway. But, again, it's actually not relevant.)
(Score: 2) by JoeMerchant on Tuesday August 01 2017, @11:19AM (7 children)
The point is: some infinities are indeed "bigger" than others, and this does occasionally arise in physics as a thing to consider.
They are all "undefined" - but, if you care to delve into the realm of "undefined" there are some relationships that still exist. They aren't as neat and clean as rational (or irrational) mathematics, but they do exist.
🌻🌻 [google.com]
(Score: 1) by khallow on Tuesday August 01 2017, @12:09PM (2 children)
You're not going to find these "bigger" infinities by dividing by zero. For example, is (1/0)/0 bigger than 1/0? No, because (1/0)/0 = 1/(0*0) = 1/0.
(Score: 3, Interesting) by JoeMerchant on Tuesday August 01 2017, @12:34PM (1 child)
See, this is why traditional mathematicians just scream "UNDEFINED, STAY OUT!" because their tools are broken on that side of the undefined line. New tools need to be defined to start getting a handle on "the other side" of undefined.
Of course, the practical applications are quite esoteric and limited in real life, but there are some physicists who think they have reasons to go there...
🌻🌻 [google.com]
(Score: 1) by khallow on Tuesday August 01 2017, @12:49PM
Already happened in the 19th century with sequences and eventually big O notation. The problem was that we got too many choices and no obvious choice for how to do it. Cardinality has its own problems, but it at least has well defined ways to find higher infinities.
(Score: 2) by Wootery on Tuesday August 01 2017, @12:55PM (3 children)
To each paragraph: citation, please.
That's an outright contradiction. If you come up with some new function that extends the division operator, well, that's all you're doing. It's not meaningful to 'delve into the realm of the undefined'. That would by definition be the same sort of nonsense as how many angels can dance on the head of a pin?
(Score: 2) by JoeMerchant on Tuesday August 01 2017, @01:05PM (2 children)
Need better definition of angels, and what constitutes their dancing...
🌻🌻 [google.com]
(Score: 2) by Wootery on Tuesday August 01 2017, @01:20PM (1 child)
Just so.
I wasn't kidding about the citations though - if you have a good source on this stuff, do please post a link.
(Score: 2) by JoeMerchant on Tuesday August 01 2017, @01:40PM
Sorry, I have hazy recollection from physics class in the 1980s... the truth is out there, but I don't have a good map.
🌻🌻 [google.com]