Mathematics and Infinity - Dr. Tony Weathers - May 3, 2012

This lecture discusses comparisons in Finite Sets and Infinite Sets, Rational Numbers, Integers, Irrational Numbers, and Infinity.

[@jodi183]

thank you for posting this. Has helped me out with my math test.

[@selvarodrigo]

Excelllent presentation. Thanks.

[@RichardAlsenz]

Thanks for posting the video. Very informative!

[@LaureanoLuna]

It's a cardinal numer: the set of all ordinals of cardinality aleph_1 (or less), which is the set of all ordinals of cardinality aleph_0 (or less), which is the set of all finite ordinals. As no set is self-membered in standard set theory, the set of all ordinals of cardinality X or less must be of cardinality greater than X.

[@666ilanam]

Excellent presentation ... deserves far more hits! But mathsphobia rules!

[@RuthBingham]

What comes immediately before 0.9999999... ?

[@TheWdayton]

You mentioned a book in the beginning but did not tell us the author. I cannot look it up.

[@grizzlyadamsmathematics]

The book in question is "Infinity" by Lillian Lieber, published by Rinehart in 1953.

[@smawzyv]

Would someone please explain Aleph 2 to me?

[@tabamal]

The problem with Cantor infinite sets and his diagonal proof is that he underestimated the complexity of natural numbers.. He disregarded the warnings of great minds, such as Aristotle, Gauss, Poincaré, etc...

[@vectorshift401]

+Ous Alam LIKE WHAT?

[@Chris-5318]

@@vectorshift401 C'mon Ous, you've had 3 years now.

[@Doppe1ganger]

This was a cool story, but I'm missing the part where you disproof Aristotle and justify using infinite sets. Unfortunately everything after this is nonsensical gibberish.

[@RichardAlsenz]

If 0 is the lack of something how can it be a number? Why isn't called a picture (or symbol) of nothing?

[@trumblewumblehumble]

0 is not a lack

[@RichardAlsenz]

Ok, then where is nowhere? Is it only in your brain? Or at the end of ad infinitum?

[@trumblewumblehumble]

Richard Alsenz nowhere is not a mathematically defined term. 0 is, first and foremost, a number. A number with special properties, but still a number. It does not, per se, stand for anything in the real world. We only tend to find the number 0 useful to describe the absence of anything in some real life applications. But that is nothing inherent to the number 0, but just a model of mathematical expressions with which we can describe a part of reality. But 0 itself can describe many things, not only absence of something. 0 degree celsius (temperature) for example is not the absence of heat, but the temperature where water freezes. You could of course use other degree measures, but in this one, 0 works just fine as an orientation point. 0 itself however, as a mathematical object, has nothing to do with anything related to real life.

[@RichardAlsenz]

0 degree celsius is typical of using 0 as a number. Of course it is not the absence of heat and it means something to you and any scientists. A common mistake does not overcome the problem that to qualify as scientific it must be measureable and repeatable. The special property of 0 is you can not measure it, it has no measureable property. If it is not measurable it is not scientific, it must be taken on faith alone. If 0 is a mathematical object then it possess irrationality at best and that has no properties as Tesla suggested. If that makes it a mathematical reality then what? You have in your mind as a conclusion the representation of nothing is a reality. You could use that to prove dark matter exists, at least at nowhere. This is why one can use the principle to prove 1/n is 0 ... if you can to ad infinitum and that can only be accomplished with faith not fact.

[@trumblewumblehumble]

Richard Alsenz That depends on what you call a measure. 0, as a number, can be the result of measuring something. For example the geometric object of a dot could be measured as 0. The special property of 0 as a number si definitely not anything related to measurements. The special property is that it is the neutral element of addition and has no inverse number in the real numbers. And those are the only special properties of 0. You make the common mistake of equating a mathematical object with things from the real world, but that never works. Math can describe the real world, but its objects exist in another realm. MAth has nothing to do with faith: If you can construct something or show that it has to exist, then it exists. And 0 can be constructed in many different ways. There are a lot of mathematical objects which cannot be measured. You cannot use zero to prove anything real life related. Zero is a number which is very thouroughly defined, as are the rings or fields of which zero is an element. 1/n is never zero and it has also nothing to do with faith. The limit of 1/n as a function or sequence is zero, that is a big difference and it just means that there is no number which is closer to zero than 1/n for every possible n. That is a well defined mathematic term, as is the real numbers. Measurement theory is an important part of calculus, and there are, for example, functions which do not have an integral value (which is one way to measure a function).

[@MisterrLi]

In your "Comparing Infinite Sets" {1,2,3,4,5,...} {2,4,6,8,10,...} (known as Galileo's paradox), what is the size relation between these sets? On the one hand, you can pair up the two sets 1 to 1. On the other hand, you can subtract all elements in the second set from the first set to end up with the odd numbers. Now, which method is the "true" one? Answer: the latter one, which also means that the set of the natural numbers is strictly smaller than the set of the rational numbers. Cantor believed in the equivalence of those sets and in the non-existence of infinitesimals, he was wrong in both cases. Proof of this can be found in the end of this comment. I'm shocked to see so many videos on youtube showing the same ignorance to precise transfinite measurement. There is a more precise way to deal with infinities: The Numerosities (replacing the less precise versions of infinite measures, the Cardinals and Ordinals of Cantor). see www.dm.unipi.it/~dinasso/papers/13.pdf

[@XnGeL92]

+MisterrLi You're saying that Cantor's set theory is wrong?, because I don't count with the knowledge for understand your attachment paper

[@MisterrLi]

+Angel Eduardo T. López Cantor's set theory and the cardinals isn't wrong, just less precise.

[@vectorshift401]

+MisterrLi Can't you also match up the sets by assigning 1 to 2, 2 to 6, 3to10, and so on then removing all the matched up numbers from both sets leaving the even numbers set with an infinite number of elements left over {4,8,...} so now the even number set would be larger by the logic above? Isn't there an infinite number of ways to do this to get various sets "left over"? Matching all the elements in both sets eliminates this ambiguity and does correspond to one aspect of what we mean by saying that two sets are equal. For finite sets differing match up schemes doesn't matter it always gives the same result for whether two sets are equal but not for infinite sets. The ability to produce a scheme that can "use up" all the elements of both sets removes that ambiguity so it seems a much more fundamental notion of equal size. The example given above doesn't illustrate at all that the set of natural numbers is in way greater then the set of even numbers.

[@MisterrLi]

Vector Shift Contrary to popular belief, you can compare countable infinite sets that are not equal in set size (one set has more elements) and say one is bigger than the other. The way you match up shouldn't matter, this is obvious with finite sets but not as clear with infinite sets. A bijection pairing of a set is a way of comparing sets in bigger and smaller infinities, but it is not a very precise measurement tool, this is where 'proper subset' comes in (the whole is greater than the part). The only way you can fail is to have badly defined infinite sets, like calling a set only 'infinite' or 'countable', which is not precise enough to use in caucluations other than as a kind of limit. Cantor used the 'bijection' as the fundamental notion of equal size. The problem with this is that it is a too imprecise measure, every countable infinite set can be matched up with the natural numbers in some way, no matter if it contains much less numbers compared to the natural numbers (that is, it is a proper subset to the natural numbers). Therefore we need a more precise measure than cardinality, and we have that with the 'numerosities'. Numerosities takes away the paradoxes that cardinal numbers gave us (if we try to solve the infinite hotel paradox using cardinals for example) and replaces them with precision. In your example above, the trick is that there are an infinite number of ways you can match elements, but if there is an easy way to see that one of the set is a proper subset of the other, as with the even numbers that is a proper subset of the whole numbers, you already have made the comparison. All you can do after that is trying to complicate matters with a different pairing up of elements, but this is changing nothing, since this new pairing is obviously the first pairing (that shows the even numbers is a proper subset and thus smaller) that is only paired up differently. You are not changing the size of the sets with a different pairing, so a more complicated pairing only confuces matters. This is based on the premise that we already know the exact infinite size of the two sets, which we know since the natural numbers and the even numbers are precisely defined. So, to sum up, the set size of the natural numbers is precisely defined and the set size of the even numbers is precisely defined as the set of the natural numbers minus the odd numbers, and therefore the even numbers is a smaller set, exactly by the number of the set of the odd numbers. You can also add or subtract the infinite sets by a finite number and end up with a set wit a different size. This is possible in that a numerosity doesn't have to have a smallest infinite size as with Cantor's numbers. Numerosities can be used for an algebra on infinite numbers, this is not possible with Cantor's cardinal numbers, see maths.york.ac.uk/www/sites/default/files/Di_Nasso-slides.pdf Numerosities is a new way of solving many paradoxes of infinite countable sets. If you want more information, look here: www.math.uni-hamburg.de/home/loewe/HiPhI/Slides/forti.pdf and: www.dm.unipi.it/~dinasso/papers/13.pdf and also: selp.apnetwork.it/sito/userfiles/alessio/file/tesi/Tesi-Bottazzi.pdf

[@paske2001]

1 = 0.9999... just proves the irrational number is bad defined.. and I KNOW how to define it well.. wait for my words world :)

[@sunsunsunh]

so?

[@paske2001]

@@sunsunsunh I failed

[@sunsunsunh]

@@paske2001 :(