Injective surjective and bijective - useful and precise words, but not ones that have quite made it into my personal dialect of english yet. Great video!
Something being undecidable does not imply that it's not true or false, but that we can't know for certain means. It could mean that we can't prove it in a finite number of steps, or it could mean that we can't know if it's true or false. It's possible that the statement is not true or false, like a pseudo-Euclidian construct without a Parallel postulate. The Parallel postulate is a degree of freedom, one can get consistent mathematic results for different choices of the postulate.
The statement is undecidable from the axioms we use. If we define a different set of axioms, then for that formal system the statement could be true or false. Gödel showed that for any formal system, there's always undecidable statements.
The video was excellent!!, i learned a lot. I feel like it has opened a new window or perspective for me to see mathematics. Thanks for your wonderful work.
This is great! But I was wondering if you will be able to make a video of how the limit is defined using epsilon delta. I am trying to learn real analysis and it’s really tough.
Thanks! I would love to do a video about limits at some point but unfortunately won't be able to for a while, still working on a few others! I'd recommend looking up The Bright side of mathematics or Bill Kinney here on CZcams. Both have really nice real analysis lectures :-) (I agree though real analysis is really tough)
Years ago I also had trouble with the definition, but you only need "mathematical language comprehension maturity" to understand it. Just re-read the definition again, let's do it with the real numbers. A limit of a function f from the reals to itself at a point (real number) x is a real number L such that for every real e > 0 there exist a real d > 0 such that if y is a real number and |y-x|
The limit is the formalization of the idea that a function gets close to a value for a range around a certain input. Ask the question: “Does the function get *this* close to the “limit” value?” (choose your epsilon) “Yes, it does, in *this* range of inputs” (and there exists a delta.) Repeat this question for smaller and smaller criteria of closeness, or tackle all possible such questions at once. (for *every* epsilon, I can find a delta.)
By the way, proving the Schröder-Bernstein theorem is surprisingly difficult. Cantor stated the theorem without proof in 1887 and in 1897Bernstein and Schröder independently published proofs. Later it was discovered that the great Richard Dedekind had found a proof in 1887 already but did not publish it. It should be called Cantor-Dedekind theorem 😛 By the way, today Dedekind is considered - with Cantor - as one of the "inventors" of set theory.
Love this so much! I've also seen the cardinality of R written as 2^(aleph_0) and I've convinced myself that that's true but I never intuitively understood it. Would love to see you tackle the "algebra of infinities" so to speak.
Basically, the idea is to show that ]0;1['s size is 2^(aleph_0) by writing the numbers in binary. You have (aleph_0) decimal places, each can be 1 or 0. then just use something like tangent to map it to R.
The reason is because the cardinality of the power set of a size of infinity is considered to be a larger infinity than the previous. An example of a power set of the set {1,2,3} is {{},{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}} and the cardinality of a power set is 2 to the power of the original set (as shown by 3 entries in the original set with 2^3 entries in the power set)
The point about the set being 'limited' to what supposed to be in it, as opposed to the content of some wider set (e.g. thinking 'all even numbers' is the set of natural number) is a mistaken perception. Part of that problem is that there is a confusion between 'counting' and the labels of the objects of the set (such as 'numbers'). The label '1', and the count of 'first element' are (can be) conceptually distinct things. This matters when explaining that the size of the 'even numbers' and the size of the whole numbers are both the same because you can count (bijectionaly) both of them. Another example is software engineers who like to start at 'zero' (also labelled '0') as their first element of their set that's an ordered list/array. Their 'forever' is 0 to (aleph.null-1) ;-)
Question: I'm not questioning the veracity of what's said, but I have a problem understanding the following: Wouldn't the size of X be distinct depending of the definition of X? So, we have a general X defined as an infinite subset of R, but we don't have more information. So, if we cannot caracterize X, it wouldn't mean that we simply lack enough information about X? Therefore, we are speaking about two completely different subsets (named X) of R and trying to force a generalization of Xsub1 and Xsub2, but Xsub1=/=Xsub2, with the two of the, being necesarrilly different by the Schröder-Bernstein Theorem presented here? For example, all rational numbers "q" are a subset of R mith the same cardinality as N, but all x: x c (0;1), x c R (sorry for the incorrect typing, I can't find the correct symbols on my keyboard) share cardinality with R but not with N.
yes… last year, it was found that there exists another set whose cardinality is betweem the cardinality of the set of all natural numbers and the cardinality of the set of all real numbers. if im remembering correctly, the cardinality of the set of all real numbers is now denoted as \aleph_2 and that new discovered set has a cardinality of \aleph_1. Quanta Magazine made a video on this. it is called "Breakthroughs in Mathematics 2021" or something like that.
I'm not sure how it can possibly be disproven if it's independent of ZFC - it must be the case that you can say "There is no cardinality between that of the reals and that of the naturals" and that will be just as consistent with ZFC as saying "There exists a cardinality between that of the reals and that of the naturals". I remember Quanta magazine did semi-recently have something about the continuum hypothesis, but it wasn't *disproving* it, it was just about a proof showing that two different ways of having a cardinality between the reals and naturals (through adding new axioms to ZFC) were equivalent or something like that.
It's known that set theory without additional axioms can't prove continuum hypothesis true or false (as long as set theory itself is consistent): there are models, such as the constructible universe, where continuum hypothesis is true; and there are models where it is false. There are some reasonable-sounding axioms which can be added to the set theory, and which imply that cardinality of the continuum is Aleph-2 (contradicting the continuum hypothesis). So in a sense, cardinality of the continuum "ought to be" Aleph-2.
@@vardhanr8177 If you mean the article 'Mathematicians Measure Infinities and Find They’re Equal', then the result was something else. There are various sets for which it's consistent that they're strictly between natural and real numbers; these sets are known as "cardinal characteristics of the continuum". (One such set is the smallest cardinality of a non-measurable set. Assuming continuum hypothesis, cardinality of the continuum is Aleph_1, and so is cardinality of the smallest non-measurable set - a countably infinite set is measurable and has measure 0; conversely, any set with positive measure has cardinality of the continuum. But suppose cardinality of the continuum is Aleph_2. Then cardinality of the smallest non-measurable set may be either Aleph_2 [i.e. all sets of cardinality Aleph_1 have measure 0], or Aleph_1 [there exists a non-measurable set of cardinality Aleph_1, i.e. with strictly lesser cardinality than the continuum].) The result involved two different sets p and t. It's known that 1) both p and t are uncountable; 2) both p and t have cardinality no greater than the continuum, and 3) cardinality of p is no greater than t. It was thought to be consistent that cardinality of p is strictly less than t; this turned not to be the case, and cardinality of p and t can be proven to be the same. (Assuming continuum hypothesis the situation is simple: because the two cardinalities are uncountable and no greater than the continuum, it follows that both are equal to Aleph_1. But without continuum hypothesis it's consistent that p and t are strictly less than the continuum.)
@@MikeRosoftJH They're referring to the article "How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer." Asperó and Schindler proved that Martin’s maximum++ implies (*). These were two competing axioms that each implied that there exists exactly one cardinality between the naturals and the reals, and thus the cardinality of the reals is aleph-two. This may seem like an arbitrary conclusion since new axioms are being introduced, but the writers of the article argue strongly to the contrary: "In addition to the continuum hypothesis, most other questions about infinite sets turn out to be independent of ZFC as well. This independence is sometimes interpreted to mean that these questions have no answer, but most set theorists see that as a profound misconception. "They believe the continuum has a precise size; we just need new tools of logic to figure out what that is. These tools will come in the form of new axioms. “The axioms do not settle these problems,” said Magidor, so “we must extend them to a richer axiom system.” It’s ZFC as a means to mathematical truth that’s lacking - not truth itself."
It's not "we might never know it". We already knew it. It's true in some set theories (like in constructible universe), but false in some other set theories.
So no matter how many times I rewatch this, I can’t quite tell how the bijective function g could possibly relate N to X, where N is all rational numbers and X is a subset of real numbers. Didn’t you say earlier that the set of real numbers between any two rational numbers is greater than the total number of rational numbers? so how could you possibly define a subset of real numbers as X and it be equal to N??
Hello, I realize your comment is five moths old so I am not sure if this is still helpful to you. I suspect the problem is you assume a subset of R to be continuous (interval), but in fact any collection of any real numbers is a subset of R. For instance, N itself is a subset of R. If X = N than surely there is no problem relating N to X via bijection :)
The idea that the question about cardinality seems innocuous is surprising to me. I don't believe the average joe with an engineering degree will have any idea on how to even start to work on it, in particular, because is for all initine subsets of R.
No, whether a statement is true or false depends entirely on the axioms you define, so if you're axioms don't make a statement uniquely determinable as true or false, the statement is neither true nor false in your mathematical system.
The |A| less than or equal to |B| and vica versa being equivalent to |A|=|B| is exactly the statement that "if there exists an injective function f from A to B and an injective function g from B to A then there exists a bijection h from A to B". You can't use the conclusion of a theorem to prove the theorem 😂
Thanks for a beautiful and clear presentation. I still have a problem with Cantor's diagonality argument, which I find arbitrary. If you look at the hypothetical table represented by Cantor, I see a fallacy in the notion of rows. There is no possibility of distinct rows in the table because the decimals are infinite. All one can say about reals is that they are uncountable and as such the concept of cardinality doesn't apply there. I'd argue also that by inference, the notion of set doesn't therefore apply either to reals. Sets, I find (pardon my resorting to intuition here) must be enumerable otherwise they are ill-defined and misconstrued for computation or logic. Another, likely unrelated idea, is that reals are only meaningful between zero and one, and that number with infinite decimals are always the combination of a natural and a real.
Maybe this way of representing it is clearer: N→R[0,1] 1→0.1 2→0.11 3→0.111 ... n→0.11111...=Σ(1/10ⁱ), from i=1 to i=n So there isn't any natural number that can be mapped to a number x=0.2 for instance. Or to 0.112, etc Edit: Although now that I think about it, it would be a bad example since those numbers above are rational and it was shown that they have the same cardinality as the natural ones. The answer would be in the irrational numbers, and that would lead to the conclusion that there are infinitely more irrational numbers than rational numbers, and that the cardinality of the real numbers is the same as that of the irrational numbers. wow. I think that with all the digits of 𝛑 and changing one digit in the next natural number and so on will be a good example. But your problem will still being the same. 😅
Hi, math major here. There any axiomatic systems in mathematics that in fact use the ideas you propose, I have definitely heard of systems attempting to get rid of uncountable sets entirely. It however turns out that these systems often lack the flexibility needed to deal with concepts such as continuity and topology. As for the diagnolization argument the table is really a simplification given to students and people unfamiliar with mathematics to help their understanding, in reality the argument propose by Cantor is much more abstract and does not rely on any counting just on the definition of counting and the integers. As for your last point about the reals only existing between 0 and 1 it is of course your right to define them that way and it's perfectly valid, that however loses some of the properties we really enjoy about the reals such as closure under addition, multiplication, subtraction. That is the main reason they are defined to be so. Let me know if I can help explain any of these concepts more specifically it's a passion of mine.
@@pablogh1204 the main problem with your proof is that you enforce an enumeration on the reals, the crux of the argument is that no enumeration can exist and so you cannot disapprove a single instance of an example enumeration and be done you need to be a lot more general.
@@yakov9ify I really appreciate you taking the time to reply to my arguments, and especially to respect them, for what they're worth. I have been at times insulted by rather dogmatic mathematicians when making this statement, and it seemed that all manner of intuition had to be banished. Intuition was core to mathematics until the turn of the 20th century and I think we need to distinguish mathematics as a language, the objects of mathematics, and the productive/predictive nature of mathematics as it pertains to either itself, or to objects of the world. I've been fascinated by Russell's tribulations, and Gödel's demonstrations. There seems to be several schools of thought about the nature of math, which from an epistemological point of view, seem to fall into the anthropic dilemma. Mathematicians are rightfully disciplined in the operational realm of the math corpus they have learned, but they seem to rarely step out of this operational realm to think about the metalanguage and the metaphysics of mathematical thought. The danger there is that one may (although this risk is not systematic) end up producing mathematical ideas that bear no mapping to the phenomenological world, only to the internal world of the mathematical 'language game', as Wittgenstein would put it. The same can be said of logic, insofar as logicians attempt to remove from it, all manner of semantics. My sentiment is that the paradox is the proper horizon of logic - if you imagine the world as an ocean, logical propositions make statements about the world that consider the infinite world from a point of view, and invariably the limit of this viewpoint is a horizon - a paradox. The horizon itself does not exist ; it is an artefact of the viewpoint, looking at a plane extending beyond the abilities of the calculus ratiocinator. Thus the idea that logical propositions can be used to decide the truth of concepts is only valid for the most basic objects - you could say 'a priori' statements. As soon as your propositions become more complex, ambiguity arises which renders the effectiveness of logic nil. I also see this as similar to the three-body problem : Perfectly rational, simple system that leads to unpredictability. When Gödel manifests this about mathematical language, he is not making a statement about the intrinsic nature of the World, and by the same token, of truth. He is only making a statement about the ability of a language to produce all proofs. Normally, this elegant demonstration should have made most mathematicians pause about the nature of their craft, but it seems that many mathematicians today still think dogmatically that maths is the World itself. Math is a largely a projection of the World. Some fundamental parts are likely perfect mappings, while others are only "valid". One has to distinguish "this appears to be true", with "this is Truth". One has to distinguish the most basic a priori in math (which necessarily must be perfect analogues of the World; aristotelian and kantian ideas such as "extent" "existence", "equality", "relation", etc) with most of the body of work, which can only really be conjectural.
@@phpn99 well I appreciate your time to reply as well, however, I think there might be a large disconnect between your understanding of mathematics and logic for that matter than the one of mathematicians and logicians. Now I am not a logician so I can only speak from my experience with my logician friends and colleagues but as far as I know the large majority of logic never tries to answer "what is true" it only tries to answer "what can be shown to be true if I think of the world to be this way". And this is a very important distinction that to my knowledge is made very early on, we think of logic as basically the highest form of a puzzle, completely abstract but only useful in and of the fact that it is fascinating. The same can be said about mathematics, at the end of the day everything in mathematics is just showing which axioms lead to which conclusions. Mathematics never claims to prove what is "true" it never claims to prove what applies to the real world. The fact that very often these ideas 'happen' to apply to the real world through the sciences is a mere coincidence that mathematics is not responsible for. PS. I understand that you find your argument very elegant but I don't necessarily agree there, there is a large quantity of unnecessarily big words to describe quite simple ideas. The beauty of a good argument comes from its simplicity not from its complexity. Either way very nice to have a discussion with somebody online rather than a virtual insult fest.
I will just comment to help you with the algorithm because you’re amazing and deserve more views!
I don't have anything terribly meaningful to say, just wanted to say thanks for making these videos (comment for "the algorithm", as the kids say)
Thanks for the support! (and feeding the algorithm lol)
Injective surjective and bijective - useful and precise words, but not ones that have quite made it into my personal dialect of english yet. Great video!
Thanks, glad it was useful!
Something being undecidable does not imply that it's not true or false, but that we can't know for certain means. It could mean that we can't prove it in a finite number of steps, or it could mean that we can't know if it's true or false. It's possible that the statement is not true or false, like a pseudo-Euclidian construct without a Parallel postulate. The Parallel postulate is a degree of freedom, one can get consistent mathematic results for different choices of the postulate.
The statement is undecidable from the axioms we use. If we define a different set of axioms, then for that formal system the statement could be true or false.
Gödel showed that for any formal system, there's always undecidable statements.
The video was excellent!!, i learned a lot. I feel like it has opened a new window or perspective for me to see mathematics. Thanks for your wonderful work.
Thank you very much. Glad you enjoyed it. I learned a lot myself while researching for this video!
This is great! But I was wondering if you will be able to make a video of how the limit is defined using epsilon delta. I am trying to learn real analysis and it’s really tough.
Thanks! I would love to do a video about limits at some point but unfortunately won't be able to for a while, still working on a few others! I'd recommend looking up The Bright side of mathematics or Bill Kinney here on CZcams. Both have really nice real analysis lectures :-) (I agree though real analysis is really tough)
Years ago I also had trouble with the definition, but you only need "mathematical language comprehension maturity" to understand it. Just re-read the definition again, let's do it with the real numbers.
A limit of a function f from the reals to itself at a point (real number) x is a real number L such that
for every real e > 0 there exist a real d > 0 such that
if y is a real number and |y-x|
The limit is the formalization of the idea that a function gets close to a value for a range around a certain input.
Ask the question: “Does the function get *this* close to the “limit” value?” (choose your epsilon)
“Yes, it does, in *this* range of inputs” (and there exists a delta.)
Repeat this question for smaller and smaller criteria of closeness, or tackle all possible such questions at once. (for *every* epsilon, I can find a delta.)
By the way, proving the Schröder-Bernstein theorem is surprisingly difficult. Cantor stated the theorem without proof in 1887 and in 1897Bernstein and Schröder independently published proofs.
Later it was discovered that the great Richard Dedekind had found a proof in 1887 already but did not publish it.
It should be called Cantor-Dedekind theorem 😛
By the way, today Dedekind is considered - with Cantor - as one of the "inventors" of set theory.
Love this so much! I've also seen the cardinality of R written as 2^(aleph_0) and I've convinced myself that that's true but I never intuitively understood it. Would love to see you tackle the "algebra of infinities" so to speak.
Basically, the idea is to show that ]0;1['s size is 2^(aleph_0) by writing the numbers in binary. You have (aleph_0) decimal places, each can be 1 or 0. then just use something like tangent to map it to R.
The reason is because the cardinality of the power set of a size of infinity is considered to be a larger infinity than the previous. An example of a power set of the set {1,2,3} is {{},{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}} and the cardinality of a power set is 2 to the power of the original set (as shown by 3 entries in the original set with 2^3 entries in the power set)
Well explained. Thank you sir!
The point about the set being 'limited' to what supposed to be in it, as opposed to the content of some wider set (e.g. thinking 'all even numbers' is the set of natural number) is a mistaken perception. Part of that problem is that there is a confusion between 'counting' and the labels of the objects of the set (such as 'numbers').
The label '1', and the count of 'first element' are (can be) conceptually distinct things. This matters when explaining that the size of the 'even numbers' and the size of the whole numbers are both the same because you can count (bijectionaly) both of them.
Another example is software engineers who like to start at 'zero' (also labelled '0') as their first element of their set that's an ordered list/array. Their 'forever' is 0 to (aleph.null-1) ;-)
Question:
I'm not questioning the veracity of what's said, but I have a problem understanding the following:
Wouldn't the size of X be distinct depending of the definition of X? So, we have a general X defined as an infinite subset of R, but we don't have more information. So, if we cannot caracterize X, it wouldn't mean that we simply lack enough information about X?
Therefore, we are speaking about two completely different subsets (named X) of R and trying to force a generalization of Xsub1 and Xsub2, but Xsub1=/=Xsub2, with the two of the, being necesarrilly different by the Schröder-Bernstein Theorem presented here?
For example, all rational numbers "q" are a subset of R mith the same cardinality as N, but all x: x c (0;1), x c R (sorry for the incorrect typing, I can't find the correct symbols on my keyboard) share cardinality with R but not with N.
I thought the continuum hypothesis was disproven, I saw something online saying that
yes… last year, it was found that there exists another set whose cardinality is betweem the cardinality of the set of all natural numbers and the cardinality of the set of all real numbers. if im remembering correctly, the cardinality of the set of all real numbers is now denoted as \aleph_2 and that new discovered set has a cardinality of \aleph_1. Quanta Magazine made a video on this. it is called "Breakthroughs in Mathematics 2021" or something like that.
I'm not sure how it can possibly be disproven if it's independent of ZFC - it must be the case that you can say "There is no cardinality between that of the reals and that of the naturals" and that will be just as consistent with ZFC as saying "There exists a cardinality between that of the reals and that of the naturals".
I remember Quanta magazine did semi-recently have something about the continuum hypothesis, but it wasn't *disproving* it, it was just about a proof showing that two different ways of having a cardinality between the reals and naturals (through adding new axioms to ZFC) were equivalent or something like that.
It's known that set theory without additional axioms can't prove continuum hypothesis true or false (as long as set theory itself is consistent): there are models, such as the constructible universe, where continuum hypothesis is true; and there are models where it is false. There are some reasonable-sounding axioms which can be added to the set theory, and which imply that cardinality of the continuum is Aleph-2 (contradicting the continuum hypothesis). So in a sense, cardinality of the continuum "ought to be" Aleph-2.
@@vardhanr8177 If you mean the article 'Mathematicians Measure Infinities and Find They’re Equal', then the result was something else. There are various sets for which it's consistent that they're strictly between natural and real numbers; these sets are known as "cardinal characteristics of the continuum". (One such set is the smallest cardinality of a non-measurable set. Assuming continuum hypothesis, cardinality of the continuum is Aleph_1, and so is cardinality of the smallest non-measurable set - a countably infinite set is measurable and has measure 0; conversely, any set with positive measure has cardinality of the continuum. But suppose cardinality of the continuum is Aleph_2. Then cardinality of the smallest non-measurable set may be either Aleph_2 [i.e. all sets of cardinality Aleph_1 have measure 0], or Aleph_1 [there exists a non-measurable set of cardinality Aleph_1, i.e. with strictly lesser cardinality than the continuum].)
The result involved two different sets p and t. It's known that 1) both p and t are uncountable; 2) both p and t have cardinality no greater than the continuum, and 3) cardinality of p is no greater than t. It was thought to be consistent that cardinality of p is strictly less than t; this turned not to be the case, and cardinality of p and t can be proven to be the same. (Assuming continuum hypothesis the situation is simple: because the two cardinalities are uncountable and no greater than the continuum, it follows that both are equal to Aleph_1. But without continuum hypothesis it's consistent that p and t are strictly less than the continuum.)
@@MikeRosoftJH They're referring to the article "How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer." Asperó and Schindler proved that Martin’s maximum++ implies (*). These were two competing axioms that each implied that there exists exactly one cardinality between the naturals and the reals, and thus the cardinality of the reals is aleph-two.
This may seem like an arbitrary conclusion since new axioms are being introduced, but the writers of the article argue strongly to the contrary:
"In addition to the continuum hypothesis, most other questions about infinite sets turn out to be independent of ZFC as well. This independence is sometimes interpreted to mean that these questions have no answer, but most set theorists see that as a profound misconception.
"They believe the continuum has a precise size; we just need new tools of logic to figure out what that is. These tools will come in the form of new axioms. “The axioms do not settle these problems,” said Magidor, so “we must extend them to a richer axiom system.” It’s ZFC as a means to mathematical truth that’s lacking - not truth itself."
Great video and easy to undertsand, thank you!! 😊
It's not "we might never know it". We already knew it. It's true in some set theories (like in constructible universe), but false in some other set theories.
But how about set theory describing V?
@@elizabethharper9081 What do you mean by set theory describing the class of all sets?
How about the set representing all possible orderings of the natural numbers?
So no matter how many times I rewatch this, I can’t quite tell how the bijective function g could possibly relate N to X, where N is all rational numbers and X is a subset of real numbers. Didn’t you say earlier that the set of real numbers between any two rational numbers is greater than the total number of rational numbers? so how could you possibly define a subset of real numbers as X and it be equal to N??
Hello, I realize your comment is five moths old so I am not sure if this is still helpful to you. I suspect the problem is you assume a subset of R to be continuous (interval), but in fact any collection of any real numbers is a subset of R. For instance, N itself is a subset of R. If X = N than surely there is no problem relating N to X via bijection :)
The idea that the question about cardinality seems innocuous is surprising to me. I don't believe the average joe with an engineering degree will have any idea on how to even start to work on it, in particular, because is for all initine subsets of R.
I think it would be a little inaccurate to say undecidable as neither true nor false, isn't it?
No, whether a statement is true or false depends entirely on the axioms you define, so if you're axioms don't make a statement uniquely determinable as true or false, the statement is neither true nor false in your mathematical system.
The |A| less than or equal to |B| and vica versa being equivalent to |A|=|B| is exactly the statement that "if there exists an injective function f from A to B and an injective function g from B to A then there exists a bijection h from A to B". You can't use the conclusion of a theorem to prove the theorem 😂
Thanks for a beautiful and clear presentation. I still have a problem with Cantor's diagonality argument, which I find arbitrary. If you look at the hypothetical table represented by Cantor, I see a fallacy in the notion of rows. There is no possibility of distinct rows in the table because the decimals are infinite. All one can say about reals is that they are uncountable and as such the concept of cardinality doesn't apply there. I'd argue also that by inference, the notion of set doesn't therefore apply either to reals. Sets, I find (pardon my resorting to intuition here) must be enumerable otherwise they are ill-defined and misconstrued for computation or logic. Another, likely unrelated idea, is that reals are only meaningful between zero and one, and that number with infinite decimals are always the combination of a natural and a real.
Maybe this way of representing it is clearer:
N→R[0,1]
1→0.1
2→0.11
3→0.111
...
n→0.11111...=Σ(1/10ⁱ), from i=1 to i=n
So there isn't any natural number that can be mapped to a number x=0.2 for instance. Or to 0.112, etc
Edit:
Although now that I think about it, it would be a bad example since those numbers above are rational and it was shown that they have the same cardinality as the natural ones.
The answer would be in the irrational numbers, and that would lead to the conclusion that there are infinitely more irrational numbers than rational numbers, and that the cardinality of the real numbers is the same as that of the irrational numbers. wow.
I think that with all the digits of 𝛑 and changing one digit in the next natural number and so on will be a good example. But your problem will still being the same. 😅
Hi, math major here. There any axiomatic systems in mathematics that in fact use the ideas you propose, I have definitely heard of systems attempting to get rid of uncountable sets entirely. It however turns out that these systems often lack the flexibility needed to deal with concepts such as continuity and topology. As for the diagnolization argument the table is really a simplification given to students and people unfamiliar with mathematics to help their understanding, in reality the argument propose by Cantor is much more abstract and does not rely on any counting just on the definition of counting and the integers. As for your last point about the reals only existing between 0 and 1 it is of course your right to define them that way and it's perfectly valid, that however loses some of the properties we really enjoy about the reals such as closure under addition, multiplication, subtraction. That is the main reason they are defined to be so. Let me know if I can help explain any of these concepts more specifically it's a passion of mine.
@@pablogh1204 the main problem with your proof is that you enforce an enumeration on the reals, the crux of the argument is that no enumeration can exist and so you cannot disapprove a single instance of an example enumeration and be done you need to be a lot more general.
@@yakov9ify I really appreciate you taking the time to reply to my arguments, and especially to respect them, for what they're worth. I have been at times insulted by rather dogmatic mathematicians when making this statement, and it seemed that all manner of intuition had to be banished. Intuition was core to mathematics until the turn of the 20th century and I think we need to distinguish mathematics as a language, the objects of mathematics, and the productive/predictive nature of mathematics as it pertains to either itself, or to objects of the world. I've been fascinated by Russell's tribulations, and Gödel's demonstrations. There seems to be several schools of thought about the nature of math, which from an epistemological point of view, seem to fall into the anthropic dilemma. Mathematicians are rightfully disciplined in the operational realm of the math corpus they have learned, but they seem to rarely step out of this operational realm to think about the metalanguage and the metaphysics of mathematical thought. The danger there is that one may (although this risk is not systematic) end up producing mathematical ideas that bear no mapping to the phenomenological world, only to the internal world of the mathematical 'language game', as Wittgenstein would put it. The same can be said of logic, insofar as logicians attempt to remove from it, all manner of semantics. My sentiment is that the paradox is the proper horizon of logic - if you imagine the world as an ocean, logical propositions make statements about the world that consider the infinite world from a point of view, and invariably the limit of this viewpoint is a horizon - a paradox. The horizon itself does not exist ; it is an artefact of the viewpoint, looking at a plane extending beyond the abilities of the calculus ratiocinator. Thus the idea that logical propositions can be used to decide the truth of concepts is only valid for the most basic objects - you could say 'a priori' statements. As soon as your propositions become more complex, ambiguity arises which renders the effectiveness of logic nil. I also see this as similar to the three-body problem : Perfectly rational, simple system that leads to unpredictability. When Gödel manifests this about mathematical language, he is not making a statement about the intrinsic nature of the World, and by the same token, of truth. He is only making a statement about the ability of a language to produce all proofs. Normally, this elegant demonstration should have made most mathematicians pause about the nature of their craft, but it seems that many mathematicians today still think dogmatically that maths is the World itself. Math is a largely a projection of the World. Some fundamental parts are likely perfect mappings, while others are only "valid". One has to distinguish "this appears to be true", with "this is Truth". One has to distinguish the most basic a priori in math (which necessarily must be perfect analogues of the World; aristotelian and kantian ideas such as "extent" "existence", "equality", "relation", etc) with most of the body of work, which can only really be conjectural.
@@phpn99 well I appreciate your time to reply as well, however, I think there might be a large disconnect between your understanding of mathematics and logic for that matter than the one of mathematicians and logicians. Now I am not a logician so I can only speak from my experience with my logician friends and colleagues but as far as I know the large majority of logic never tries to answer "what is true" it only tries to answer "what can be shown to be true if I think of the world to be this way". And this is a very important distinction that to my knowledge is made very early on, we think of logic as basically the highest form of a puzzle, completely abstract but only useful in and of the fact that it is fascinating. The same can be said about mathematics, at the end of the day everything in mathematics is just showing which axioms lead to which conclusions. Mathematics never claims to prove what is "true" it never claims to prove what applies to the real world. The fact that very often these ideas 'happen' to apply to the real world through the sciences is a mere coincidence that mathematics is not responsible for.
PS. I understand that you find your argument very elegant but I don't necessarily agree there, there is a large quantity of unnecessarily big words to describe quite simple ideas. The beauty of a good argument comes from its simplicity not from its complexity. Either way very nice to have a discussion with somebody online rather than a virtual insult fest.