Comments by "MC116" (@angelmendez-rivera351) on "How to Count" video.

That is because you are getting impatient. You have to wait until after he has discussed counting before we can get to the topic of integers. Just relax.
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@johnwallace918 I think the short answer here is that arrays are not merely sets, but sets with additional structure.
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@Happy_Abe No, that does not use the definition of a 2-tuple. A 2-tuple is itself a function, a function from 2 to the target set, where 2 = {0, 1}. The object {{x}, {x, y}} is actually called a Kuratowski pair. The corresponding two-tuple is instead the set {{{0}, {0, x}}, {{1}, {1, y}}}, which is built from the Kuratowski pairs {{0}, {0, x}} and {{1}, {1, y}}. The distinction is subtle, but it exists, because there is no 3-element analogue for the Kuratowski pair construction, while there is such a thing as a 3-tuple: a function from 3 to the target set, where 3 = {0, 1, 2}. In this case, the three tuple would look like {{0}, {0, x}}, {{1}, {1, y}}, {{2}, {2, z}}}.
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@orisegel4055 You need set theory in order to explain what you mean by a mathematical statement,... How so?
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@orisegel4055 Zermelo-Fraenkel set theory uses first-order logic. How is set theory necessary to explain first-order logic?
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Regarding Galileo's paradox, you said that a function which matches every possible integer which is an even number with itself is injective. Fair enough, more integers than even numbers. But what about the follow-up: you have a function where each integer is assigned to its double, and it is a bijective function. How is that possible? I do not understand your question, but hopefully, what I will say is still helpful. The set of integers Z has a proper subset that we call the set of even integers, which we denote as 2Z. There are two functions that were mentioned in the video from 2Z to Z. The former function, f, is the function such that f(n) = n for all even integers n. f is an injection, but not a surjection, so it is not a bijection. The latter function, g, is the function such that g(n) = 2·n = n + n. g is an injection AND a surjection, so it is a bijection. The question that is important here is, do there exist any bijections from 2Z to Z at all? If the answer is no, then 2Z is smaller than Z, since an injection between 2Z to Z does exist. If the answer is yes, then 2Z is the same size as Z. So, what is the answer? The answer is yes: there exists at least one bijection from 2Z to Z, and g is one of those bijections (there are infinitely many such bijections, actually, g is just the simplest one to talk about). Actually, this gives us a hint as to why we denote the set of even integers as 2Z to begin with. If there are more integers than even integers, how is the second case showing a bijection and not an injection as well. It is not the case that there are more integers than even integers. I believe you are misunderstanding what the video is saying. Intuitively, one may think there are more integers than even integers, based on the fact that f is not a bijection, but g is a bijection, so despite what your intuition would say, 2Z and Z have the same size.
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The object you described is not a set. The "set" you described cannot exist thanks to the axiom of regularity. The axiom of regularity states that for every nonempty set x, there exists a set y that has no elements that also belong to x. So there is no set U such that all sets are elements of U.
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@zhadoomzx Yes
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A bijection is a surjection and an injection. So if f is a bijection from x to y, then it is an injection x to y, hence |x| =< |y|, and it is also a surjection from x to y, hence |x| >= |y|, and therefore, |x| = |y|.
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Your questions are answered by the video itself.
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He was never trying to prove induction. I suggest you rewatch the video, and this time with the intent of actually understanding what he is trying to do.
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@zenithparsec He does not have to prove induction. I am sorry you do not understand
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The ordered pair (2, 1) would not be {{2, 1}, {1, 2}}. The definition you provided does not support your conclusion. The only weird aspect of the definition (a, b) = {{a}, {a, b}} occurs when a = b, but even this is not actually very weird: while you do get {{a}} in that case, this is not an issue, because a is never equal to {a}.
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@rmsgrey Fair enough. I was just confused because I did not realize OP was talking about the video's definition, but the other one.
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People don't seem to talk about mathematics this way. People do not talk about mathematics that way, because almost none of the things you mentioned are true of mathematics. The descriptions you provided describe science, not mathematics. In fact, it is impossible for those descriptions to be true of mathematics, because the description regarding statistics requires mathematics in an epistemically a priori manner. Before you can philosophize and determine that statistical analysis is an important component of the scientific method, you must first know what statistical analysis even is, and to know what statistical analysis even is, you must learn about the foundations of mathematics. So if your description of science is correct (and I argue that it is), then mathematically is necessarily epistemically prior to the scientific method. The reason you are getting confused is because you are treating the philosophy of science as if it were equivalent to the philosophy of mathematics. It is not.
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Is there any sense in which one set is still 'bigger' (scaled) compared to the other? The problem here is that you are conflating "scaled" and "bigger," as if they mean the same thing. They absolutely do not mean the same thing. In fact, the set 2Z is indeed a scalar multiple of Z, in the Minkowski scalar multiplication sense. For more information on what this means, you can read the Wikipedia article on "Minkowski sum." However, it is still true that |2Z| = |Z|. The problem that you are neglecting is that if you multiply |Z| by 2, you still get |Z|. Is this a problem with the notion of being able to 'count infinity'? No. What this is, it is a defining characteristic of infinite sets. Your intuition tells you that if A is a proper subset of B, then A is smaller than B. This is only true if A and B are finite sets. If you informally treat the cardinality of a set as a function output, with the set being the input, then you can view this as saying that the cardinality function is monotonic with respect to subsethood, but not strictly monotonic, even though your intuition is telling you that it must be strictly monotonic. Of course, this is just an analogy: cardinality is not a proper function. I suspect that something along this path is the break that allows the axiom of choice to lead to multiple copies of an object. No, not at all. Even if the axiom of choice is false, all countable sets are still the same size, even one is a proper subset of another. The axiom of choice has no bearing on that. What the axiom of choice does is it guarantees that the cardinalities of sets can be well-ordered, meaning that for every size of a set, there is always a well-defined "next size" or "next cardinality," and that if you have a set of sizes, that there is always a smallest size in that set of sizes.
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@tBagley43 Yes, you are right, but Fullfungo's qualm is that such a proof was absent from the video.
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Yes, you would have to have already constructed the set of integers from the set-theoretic axioms. The proof assumes we already have done that, which admittedly, is an anachronism, but since we know such a construction exists anyway, this is not an issue worth worrying about.
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@Lukasek_Grubasek Formal set-theoretic definitions for addition and multiplication for the natural numbers do exist, and they extend to the integers, the rational numbers, and other sets quite naturally. As to whether these definitions are necessary: if you want to define the integers set-theoretically, then you definitely need to define addition beforehand. Multiplication is optional, though, and 2·n would, in this case, be merely a shorthand for n + n, rather than an actual product. However, to define the rational numbers, you absolutely must define integer multiplication first.
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The former way does not get messy even if a = b, since a = {a} is false for all sets a.
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If f maps x to y, and f maps x to z, then y = z. This is the rigorous definition.
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However, the set of all even integers is technically still smaller than the set of all integers, because given any hypothetical "final" integer as the ultimate max value, that value could never be doubled, and so would not have a match in the "even" set. This objection can be dismissed, because there is no hypothetical final integer. There is no largest or smallest integer, not even hypothetically. A bijection from the set of all integers to the set of all even integerd exists. Period. There is nothing to be said here. If you claim no such a bijection exists, then you must prove it.
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It is not, and the ending of the video literally explains why this is.
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This causes issues when mapping the element {a} to a.
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@tBagley43 I do not believe A = {{A}} is possible.
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@tBagley43 It is unclear to me that what you say is true. If A = {{A}}, then A = {{{{A}}}}, and so A = {{{{{{{{A}}}}}}}}, and really, you can have the number of braces an arbitrary power of 2. But if this is the case, then it seems to me A is ill-defined. In any case, it does not seem to me that the set A such that A = {{A}} can be constructed from the axioms.
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A relation is a function, but without the strict requirement that an element of the domain be related to exactly one element from the codomain.
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@reddy1936 He did not construct integers either, he constructed the natural numbers. The integers are constructed differently.
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@kruksog It actually is not that difficult if you are using order-theoretic concepts.
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Possibly, but that would not happen in the near future, since we are still at the stage where we are discussing counting.
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Did you not watch the rest of the video?
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@Happy_Abe No, that does not use the definition of a 2-tuple. A 2-tuple is itself a function, a function from 2 to the target set, where 2 = {0, 1}. The object {{x}, {x, y}} is actually called a Kuratowski pair. The corresponding two-tuple is instead the set {{{0}, {0, x}}, {{1}, {1, y}}}, which is built from the Kuratowski pairs {{0}, {0, x}} and {{1}, {1, y}}. The distinction is subtle, but it exists, because there is no 3-element analogue for the Kuratowski pair construction, while there is such a thing as a 3-tuple: a function from 3 to the target set, where 3 = {0, 1, 2}. In this case, the three tuple would look like {{0}, {0, x}}, {{1}, {1, y}}, {{2}, {2, z}}}.
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@Trequetrum8 The axiom of regularity is one of multiple axioms that can help you prove that for all, A is not an element of A. It just happens to be one preferred by mathematicians, but other choices exist in other set theories.
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@Happy_Abe Wikipedia, as usual, is oversimplifying things, since it is not meant to be treated as an introductory textbook to the topic. A Kuratowski pair of x, y, namely {{x}, {x, y}}, is different from a 2-tuple of x, y, namely 2 —> {x, y}.
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@martinepstein9826 Yes, the Kuratowski definition is what seems to be the best definition, with the least amount of difficulties and inconvenient situations as a consequence.
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@Trequetrum8 I apologize. I believe I never actually answered the question you asked me. The axiom of regularity is also called the axiom of foundation, because it implies that set membership, informally, is a well-partial order on the universe of sets, which means that it is not possible for cyclical membership to occur: you cannot have that A is in B and B is in A, hence you cannot have B = {{B}}, or B = {{{B}}}, or anything of the sort.
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I mean, the reason their way of counting makes as much sense as it does is because ultimately, they are just verbalizing the fact that 11 = 10 + 1, 25 = 2·10 + 5, 345 = 3·100 + 4·10 + 5, which is ultimately how decimal notation is constructed.
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In set theory, "encoded" and "defined" are synonymous. Hence, your proposition is false.
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@annaclarafenyo8185 The point is that the set theoretic encoding isn't any better than, say, a type theory encoding, or just the Peano axioms. This is false. Set theory is a general theory of sets, while the Peano axioms only endow structure on one specific primitive notion, for which a specific corresponding set does exist in set theory. There's nothing special about the set-theoretic embedding to allow it to give deeper insight into the nature of counting. This is also false. What set-theoretic formalism does is it allows you to study combinatorics without relying on number theory as much, and it also extends the idea of counting to the more general idea of cardinality, which is an important one in many areas of mathematics. Counting is a very simple thing, and set-theoretic definitions of it are kind of mismatched with the idea. This is solely your opinion, and not an opinion that corresponds to reality.
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