# ➊ Comparing RussellВґs Paradox, Cantors Diagonal Argument And

Wednesday, September 29, 2021 2:13:48 PM Cantor's Infinity Paradox - Set Theory

This is a proof by contradiction that the assumption is actually false! That is, the set of all sequences is not countable. By definition, a sequence can only contain countably many elements. Cantor's diagonalization argument was taken as a symptom of underlying inconsistencies - this is what debunked the assumption that all infinite sets are the same size. The other option was to assert that the constructed sequence isn't a sequence for some reason; but that seems like a much more fundamental notion. In general, we prefer to question axioms rather than definitions, wherever possible, because definitions are far more fundamental.

Also, nothing becomes a "paradox". Russell's paradox prompted the creation of modern set theory, in which the "paradox" is just a theorem. The "twin paradox" in relativity is just a thought experiment that informs the experimenter about the effects of relativity. Schrodinger's Cat, originally intended as a paradox, is now taken as an illustration of the effects of quantum mechanics. Given the option, no scientist in any field will take an example as an actual paradox; they will alter their assumptions just enough to make it no longer a paradox, rather than throwing everything out and starting fresh. In this case, the solutions to Cantor's "paradox" were to either conclude that some infinities are bigger than others, or that the definition of a "sequence" is inherently flawed in some unknown way.

The first option was clearly simpler, so we chose that. You already got a lot of answers about your errors. But let me add what Cantor's argument actually says. The most famous application of Cantor's diagonal element, showing that there are more reals than natural numbers, works by representing the real numbers as digit strings, that is, maps from the natural numbers to the set of digits. Note that there are always two sets involved in Cantor's diagonal element. You are talking about sequences. Sequences are, bu definition, functions from the natural numbers to whatever objects you form sequences of.

I know that others have already given counterexamples to this, you could actually come up with counterexamples for quite a few infinite sets. For example, the set of all real numbers between 5 and 10 is clearly uncountably infinite, but it's still a proper subset of the real numbers. You may want to research the Infinite Hotel Paradox. Suppose that you have a hotel with a countably infinite number of rooms, all of which are booked. One night, a bus pulls up with an infinite number of numbers looking for rooms. The paradoxical thing: the hotel can accommodate all of them in spite of being at "full occupancy. One point to emphasize here from the previous paragraph is that, when reasoning about infinite sets, your intuition can steer you badly wrong if you're not careful.

A lot of people accidentally import assumptions about how finite sets work and then assume that infinite sets will work the same way, and there's no requirement that they will. Therefore the problem arises. If you look into the Cantor's theorem you will see that the numbering of the sequences matters. Therefore the countability of the set T is first assumed and then after the contradiction is reached, all the blame is put in on our single assumption which is the countability. This has to be true;". It has to be true because that is how we defined it; not for the arguments you give which are naively wrong. But that's irrelevant. Hold on! And that is the entire point. So there is not paradox because we didn't do anything.

Which is Thomas Andrew's answer is the best, but I thought it is easier to swallow when we make things extremely concrete, so here goes:. Instead of 'sequences' we take Sequences of binary digits , and we add an ornamental '0. Also, the elements already have an ordering on them. Now, to paraphrase fleablood above: 'Your move. Regardless of whether or not we assume the set is countable You do realize that you just defined T to be countable, don't you? You associated every member of T with a natural number n, which is the definition of "countable.

Cantor's proof is often misrepresented. He assumes only that 1 T is the set of all binary strings, and that 2 S is a subset of T; whether it is proper or improper is not addressed by this assumption. Let A be the statement "S is countable," and B be the statement "S is equal to T; that is, an improper subset. Diagonalization proves the lemma "If A, then not B" by exploiting the way S can be counted to produce a string that is in T, but not in S. 