Is there a standard "canonical" set of axioms you're going with for this series? I know that in most axiomatic systems, one can often interchange some of the axioms with some of the most basic theorems and get an equivalent axiomatic system, so how do you decide which basic things you're going to call axioms in this series? For example, I notice you have the axiom of choice as axiom #9, but as I understand it, you could have chosen Zorn's lemma instead for that one.
Good question, I'm going with one of the versions of Zermelo-Fraenkel. ZF is the most known choice as an axiomatic set theory because it is simple and powerful. There are some possible small variations inside ZF, I'm going with the version where the axiom of the empty set is a consequence of the other axioms (specifically of the axiom of infinity), this is just a matter of personal taste, I like minimalism. The axiom of Pairs can also be avoided because it is a conaequence of the axiom of powers but many prefer to leave it in the basic set because it is just so basic ... So yes, ZF is the standard and the various versions are all equivalent. Then there are other systems of axioms ... this could be a good idea for a future video. Ciao, thx.
uh no. The first axiom of math is that contradiction is forbidden. It is in fact the only axiom of math, but you could argue that it practically requires a second axiom where-in definitions must have precise agreement before evaluation. This video is a testament to how easily a person can confuse themselves when they don't understand these axioms. They are so blatantly taken for granted that this video goes straight to the idea of a proof without noticing that the definition of a proof itself relies first on excluding contradiction. The example given to random people of balls in a jar exploit the 2nd axiom - that definitions must be agreed to precisely in advance. The presenter merely asserts without agreement that the position of the balls for example is irrelevant - that is not true in every case. As a result, the entire video is an exercise in absurdity. However, once the presenter decides to describe the balls as a mathematical set, through social contract one can reasonably argue the definition has been ascribed to exclude position as irrelevant. It is incorrect to try to force it or assume it however and that is a dramatically bad blunder. Importantly, math is a symbolic abstraction just like any other language. Therefore the first example 1+1=2 has already been rendered into the proper symbology and the balls have not; they invoke a physical reality and certainly have not been defined into a widely known symbolic abstraction fit for a true/false evaluation. Finally though, no - set theory is obviously not the foundation of math neither are its axioms. The axioms of set theory are just the axioms of set theory. The axioms of category theory are just the axioms of category theory. If you want to argue either is "the foundation of math" you would be on equal footing with either set theory or category theory. The resulting contradiction necessarily excludes both as not the foundation of math.
Hi, what you say is really interesting but "contradiction is forbidden" is not an axiom of math is a logic rule that precedes the axioms of math. Math can be constructed in various ways but the most common way is to base everything on the axioms of sets.
@@guzmat There should probably be a distinction between "the most common way" and "THE way." Wouldn't this count as an appeal to popularity? A logical fallacy? Contradiction being forbidden reality is a more fundamental truth of math than any given set of axioms. A given set of axioms define what kind of math you're working with, but you could still call what you're doing "math" even if you completely throw out every known axiom and start with a set that you've entirely made up and seem nonsensical, but are nonetheless consistent with each other.
This reply from overSquare is really interesting. I actually just paused the video right after the encounter with the two girls feeling like his explanation to them was quite unsatisfying, missing some important points and hand-wavy. They just nodded their heads in agreement at the end, but really came away not understanding anything more than when he first asked them the question. Anyway, I stumbled upon this reply from overSquare, and (having thought myself as understanding what axioms were quite well) realized that I hadn't quite considered this point about logic from the point of view of "axiomatic". It's an excellent point. (Plus I know nothing about category theory - maybe time for some RUclips searching on the subject.) Maybe logic is meta-axiomatic? Without it the universe ceases to exist kind of thing? I do recall similar thoughts when first learning about axioms in my 1st year Analysis class (from Spivak's "Calculus" at U of Toronto ~1983) and learning about proofs. I realized that we skimmed over some basic assumptions about constructing proofs based on the rules of logic. For example, why can you go from statement A to statement B in an argument? I suppose I was questioning those simple things because I was sitting among a bunch of kids who went to math-camp, but I had just transferred into this stream from a artsy film-production degree, so was learning everything cold and questioning everything that confused me. Please don't get me wrong, I greatly appreciate the video that GuzMat has created, and take my hat off to ANYONE who puts in such a strong effort to produce video that are both educational and entertaining of which his are. I just wanted to weigh in here with some more thoughts for what they're worth.
Thanks for you comment and your remarks ... they are actually really interesting and could well deserve a video on its own ... I'm not much into logic per sè but I'll think about this possibility ... ciao!
Most of the proposed foundations of math (including set theory and HoTT) are not equivalent to one another though, and they're not really trying to be. They're typically very different paradigms at the ground level. Like, no, you can't "prove" category theory from set theory or vice-versa (and frankly if you could, that would be a compelling reason to say that they independently are in fact both valid foundations, rather than that neither is as you say). For instance in set theory we prove relative consistency results (since absolute consistency results are provably unprovable), while in HoTT we use the programs-as-proofs interpretation of logic with an abstract notion of homotopy equivalence. They're not really comparable along the lines of "validity."
Why is there a second "if and only if" in the first axiom? Wouldn't an "and" be enough? C has to be in A "and" also be in B for every C. Btw. since all nine axioms listet in the video are actually the axioms of ZFC set theory, it would be interesting to hear why this perticular set theory is so special that it can be used for math as a hole. There are for example other set theories that for example have no axiom of foundation.
hi, thx ... an element C is in A "if and only if" it is in B ... or "for every C" we have that "C is in A and in B" or "C is not A and not B" ... Yes, very good question about ZFC ... I'm gonna talk more about the other axioms and the other possible axiomatization ... but I didn't want to make things too hard just from the start ... we need the time to get familiar with some concepts ... I'm gonna talk about that in the next videos ... thank you very much for you comment
Mate I get what you’re trying to say but I gotta slightly disagree with you. The axiom you name is this video is more of a definition of what it means for 2 sets to be equal. Also, the jar thing, we have no idea whether the elements of a jar are ordered or have specific coordinates, so the equality symbol there needs extra definition. But I appreciate the attempt to teach rigorous logic to the world.
Ah, very nice 😘. By the way, thank you for covering axioms in mathematics. They are so widely unknown!!! I love the 1+1 = 2 question ❤️ almost everybody falsely thinks is an axiom. I disagree with your statement “you can trace the entirety of maths back to these nine axioms”. Think of what Gödel proved was never possible 🤓! Neither are these nine set-theoretical axioms the only ones in mathematics.
yes, you are right ... it is not entirely accurate ... it was just a way to introduce the subject ... we'll be talking about Godel Theorems later in the series ... thanks for you comment!!!
**ATTN: 😌😉This comment is equal to a rant from a novice mathematician who desperately wants and LONGS to know and understand mathematics.😔😉😁** Question: if math and the basic foundations for math are so hard for people to comprehend-as your video has displayed-could it be the mathematical: i.e., theorems and axioms; system being taught is flawed??? Honestly, using logic, it is a known (“proven”) fact that a vast majority of people, globally, have issues trying to understand mathematics as they have been taught. Therefore, could it be the commonly used Greek adapted mathematical, now called the Algebraic system, is grossly flawed??? How many more centuries must mankind endure this dated, and, in my opinion, flawed and grossly hard to comprehend Greek philosophical interpreted version of ancient (Egyptian and Babylonian) “earth measurements,” which is now commonly known as the globally accepted Grecian term, geometry (Merriam-Webster: Geometry- Etymology…. from Greek geōmetria, from geōmetrein to measure the earth)?🤔🧮 +🧐📚👨💻=😁🧑🏻🎓!
Hi, thanks for you comment. First of all, are axioms really so hard to understand? I'm not sure, maybe some of them are. But in any case the answer to your question is: yes, math could be actually flawed and this in itself is actually a proven theorem; not kidding.
@@guzmat, Aioxms might not be hard. However, as your video shows, math is not easily learned by many. Therefore, could it be the system is flawed? If so many people cannot digest, comprehend, and retain mathematics, we must realize reality. Certainly masses of people are not flawed. Instead, we must examine mathematics for there must be something flawed in the way mathematics are taught and interpreted. This is a global issue, so maybe other cultural forms of mathematics should be taught as an alternative option. Who knows, maybe there are more mathematicians in our world but they are not able to digest the Greek interpretation of mathematics.😉
Math can probably be taught better ... and it would be really interesting to discover different and new ways to do math. How do aliens do math? Is alien math the same as ours?
This whole business of axioms makes no sense to me as it relates to logic in mathematics. Consider, the statement 1+1=2 is almost tautological and true by definition. There is no reason for an axiom because the statement cannot be untrue. This is unequivocal. That the concept of 1 can be conceptualized in consequence of the perception of “a something by itself” and that label “1” arbitrarily formulated and applied to that concept means that that which was perceived is “1” something by itself. As long as all in witness agree that “1” means a something by itself, there is no need for proof. It cannot be otherwise. That another “1” or something by itself can be perceived and understood as such is by definition possible or the original something by itself could not have been perceived, which it was (and we named it). By extension then, we cannot but understand that 1 and (or +) 1 is (1 and 1) or “2”, a second concept we formulate and name. The label 2 meaning a something by itself and another something by itself is thus by definition true. Again, no need for axioms or proof of its truth for were we unable to comprehend its unequivocal truth we could not have expressed the proposition to begin with which we did. Again, by extension and the application of the same logic, we know that 2+2=4 by the same logic which we would be obliged to apply. The further extension of these simple propositions would be by the same logic, by definition true, unequivocally. What then is the point of these extensive, sometimes redundant axioms by which sophistry like that of Goedel’s claim of the “hole” in the logic of math can be imposed. Any proposition which is true by definition requires no proof and so the insistence on the imposition of some analogously wordy and thus, unnecessary scheme is actually defiant of the logic it is purported to support or validate. Any thoughts?
Well, it depends on what you mean by congruence ... Equality is having the same elements ... A congruence relation is any relation that is reflexive (a R a), symmetric (a R b implies b R a), and transitive (a R b, b R c implies a R c) ... So in general congruence is more general than equality.
@@guzmat Equality describes the boundary, congruence fills to the boundary. The universe exists through its attempt to fulfill the Pythagorean relation, to find that integer where the hypotenuse is equal to B+3, although the relation is always congruent without integer satisfaction.
thanks for writing the comment, some of the next axioms will be gropued in a single video ... but there are some that in my opinion need a video ... because the subject becomes complex very fast ... for example there are various ways to enunciate the second axiom and the equivalence of these different statements is really tricky. I think that some time is needed to familiarise with the concepts ... but I get what you mean and I'll try to be more concise.
1+1=2 is axiomatic per Peano's Axioms. the alleged proof which appears in Principia Mathematica is a circular application of this, and thus is unsound. if 1+1=2 then there would be no need for algebra, and unit conversions would be impossible. consider, for instance: 1 foot + 1 yard = 48 inches 1 dog + 1 quail = 6 legs 1C water + 1C dirt = some mud 1 frog + 1 pond = 1 pond you can thus soundly prove that a pair of vectors with comensurable units and magnitudes of 1 sum to one vector in the same unit with a magnitude of 2. but the number 1 cannot actually even be added to anything, and attempting to do so won't yield a number, let alone the number 2.
you can even see this in the orthography. 4-3 = 1; this is a subtraction, right? -3+4=1; hang on, where did the + come from? if what we're actually doing is using positive and negative vectors, like +4, -3, and +1, which have commensurable units, then it all makes sense. however, this also means that + and - are not operators, but partial names of the unit of these vectors. further, the grammar of the orthography we're using permits us to not write the + if it occurs in initial position within an expression. that doesn't mean it's not really there, it just means we're not explicitly saying it. such rules are quite common in human languages, actually. now, reconsider the current foundation of mathematics armed with this understanding. nearly everything built up over the past two centuries assumes that + and - are mathematical operations which operate over bare numbers. but clearly this must be wrong. as such... what the hell are you doing?
1 + 1 = 2 because x + 0 = x, x + y = succ(x + pred(y)), pred(a) = b if succ(b) = a, succ(0) = 1 so pred(1) = 0, x + 1 = succ(x + pred(1)) = succ(x + 0) = succ(x), succ(1) = 2, therefore 1 + 1 = 2. Of course, the fact that succ(0) = 1 and succ(1) = 2 are logically dubious, and defining them as anything else would still be completely consistent with math, but would break this "proof". Really most of this "proof" is highly dependent on the exact definitions of "0", "1", "2", "succ", and "pred", which is where the question of "what is an axiom" starts to become less about math and more about language.
Okay look at your phone it goes from one to zero that's is all the numbers you need 1=abc 2=def all the way to 9=XYZ O stands alone (10) so if I was to SPELL something with numbers IGo by the alphabets or how equal they are spaced lets say 1to 9 =8 example 1+8=9 or 9-1=8
your videos looks like that old tv shows for children with interviews
great!
Thanks! 😃
Math for me began with my fingers.
see my last video ...
Will do. @@guzmat
The set of elements of type extended fingers :)
and it is interesting fingers are also called 'digits'. Math is the language of the entire universe.
@@GungaGaLunga777 Nice one!! Thanks for that. I dig it--get it??
This video was super engaging, thank you! The concept of axioms is so cool
Glad you liked it!
As a self taught. Finally I found the answers I am looking for
Muchas gracias
Hola si hablas en español mi canal en español tiene mucho más...gracias por tu comentario! chau!!
Keep it up, I'm waiting!
next video in the series is almost ready ... coming soon ... two weeks I think,
in the mean time I'll post some nice puzzles and shorts ...
As soon as I saw this guys face I figured he was about to tell me that I never had the makings of a varsity athlete
Don't forget Terrence Howard's axiom that states : 1x1 = 2
One times one is one squared: two dimensional.
Is there a standard "canonical" set of axioms you're going with for this series? I know that in most axiomatic systems, one can often interchange some of the axioms with some of the most basic theorems and get an equivalent axiomatic system, so how do you decide which basic things you're going to call axioms in this series? For example, I notice you have the axiom of choice as axiom #9, but as I understand it, you could have chosen Zorn's lemma instead for that one.
Good question, I'm going with one of the versions of Zermelo-Fraenkel. ZF is the most known choice as an axiomatic set theory because it is simple and powerful. There are some possible small variations inside ZF, I'm going with the version where the axiom of the empty set is a consequence of the other axioms (specifically of the axiom of infinity), this is just a matter of personal taste, I like minimalism. The axiom of Pairs can also be avoided because it is a conaequence of the axiom of powers but many prefer to leave it in the basic set because it is just so basic ... So yes, ZF is the standard and the various versions are all equivalent.
Then there are other systems of axioms ... this could be a good idea for a future video. Ciao, thx.
It's possible to prove that 1 + 1 = 2 from simpler axioms, but is it possible to do the same for successor(1) = 2?
Well, you also need the axioms to define the numbers ...
I'd say that this is true by definition: 1 is defined as a successor of 0, and 2 as a successor of 1. (I.e. 2=S(S(0)).)
nice video man
Cool video, very informative and clear
I like this channel
uh no. The first axiom of math is that contradiction is forbidden. It is in fact the only axiom of math, but you could argue that it practically requires a second axiom where-in definitions must have precise agreement before evaluation. This video is a testament to how easily a person can confuse themselves when they don't understand these axioms.
They are so blatantly taken for granted that this video goes straight to the idea of a proof without noticing that the definition of a proof itself relies first on excluding contradiction. The example given to random people of balls in a jar exploit the 2nd axiom - that definitions must be agreed to precisely in advance. The presenter merely asserts without agreement that the position of the balls for example is irrelevant - that is not true in every case. As a result, the entire video is an exercise in absurdity.
However, once the presenter decides to describe the balls as a mathematical set, through social contract one can reasonably argue the definition has been ascribed to exclude position as irrelevant. It is incorrect to try to force it or assume it however and that is a dramatically bad blunder.
Importantly, math is a symbolic abstraction just like any other language. Therefore the first example 1+1=2 has already been rendered into the proper symbology and the balls have not; they invoke a physical reality and certainly have not been defined into a widely known symbolic abstraction fit for a true/false evaluation. Finally though, no - set theory is obviously not the foundation of math neither are its axioms. The axioms of set theory are just the axioms of set theory. The axioms of category theory are just the axioms of category theory. If you want to argue either is "the foundation of math" you would be on equal footing with either set theory or category theory. The resulting contradiction necessarily excludes both as not the foundation of math.
Hi, what you say is really interesting but "contradiction is forbidden" is not an axiom of math is a logic rule that precedes the axioms of math. Math can be constructed in various ways but the most common way is to base everything on the axioms of sets.
@@guzmat There should probably be a distinction between "the most common way" and "THE way." Wouldn't this count as an appeal to popularity? A logical fallacy?
Contradiction being forbidden reality is a more fundamental truth of math than any given set of axioms. A given set of axioms define what kind of math you're working with, but you could still call what you're doing "math" even if you completely throw out every known axiom and start with a set that you've entirely made up and seem nonsensical, but are nonetheless consistent with each other.
This reply from overSquare is really interesting. I actually just paused the video right after the encounter with the two girls feeling like his explanation to them was quite unsatisfying, missing some important points and hand-wavy. They just nodded their heads in agreement at the end, but really came away not understanding anything more than when he first asked them the question.
Anyway, I stumbled upon this reply from overSquare, and (having thought myself as understanding what axioms were quite well) realized that I hadn't quite considered this point about logic from the point of view of "axiomatic". It's an excellent point. (Plus I know nothing about category theory - maybe time for some RUclips searching on the subject.) Maybe logic is meta-axiomatic? Without it the universe ceases to exist kind of thing?
I do recall similar thoughts when first learning about axioms in my 1st year Analysis class (from Spivak's "Calculus" at U of Toronto ~1983) and learning about proofs. I realized that we skimmed over some basic assumptions about constructing proofs based on the rules of logic. For example, why can you go from statement A to statement B in an argument? I suppose I was questioning those simple things because I was sitting among a bunch of kids who went to math-camp, but I had just transferred into this stream from a artsy film-production degree, so was learning everything cold and questioning everything that confused me.
Please don't get me wrong, I greatly appreciate the video that GuzMat has created, and take my hat off to ANYONE who puts in such a strong effort to produce video that are both educational and entertaining of which his are. I just wanted to weigh in here with some more thoughts for what they're worth.
Thanks for you comment and your remarks ... they are actually really interesting and could well deserve a video on its own ... I'm not much into logic per sè but I'll think about this possibility ... ciao!
Most of the proposed foundations of math (including set theory and HoTT) are not equivalent to one another though, and they're not really trying to be. They're typically very different paradigms at the ground level. Like, no, you can't "prove" category theory from set theory or vice-versa (and frankly if you could, that would be a compelling reason to say that they independently are in fact both valid foundations, rather than that neither is as you say). For instance in set theory we prove relative consistency results (since absolute consistency results are provably unprovable), while in HoTT we use the programs-as-proofs interpretation of logic with an abstract notion of homotopy equivalence. They're not really comparable along the lines of "validity."
If the two jars are separated in space, they are differently located.
You are a most excellent teacher.
thank you!!!
axiom is what we can't do any better
Awesome video. What's the name of the city @3:04? I absolutely adore the buildings behind those people.
Hi, that's Florence (Firenze) in Italy, ciao!!!
Why is there a second "if and only if" in the first axiom? Wouldn't an "and" be enough? C has to be in A "and" also be in B for every C. Btw. since all nine axioms listet in the video are actually the axioms of ZFC set theory, it would be interesting to hear why this perticular set theory is so special that it can be used for math as a hole. There are for example other set theories that for example have no axiom of foundation.
hi, thx ... an element C is in A "if and only if" it is in B ... or "for every C" we have that "C is in A and in B" or "C is not A and not B" ...
Yes, very good question about ZFC ... I'm gonna talk more about the other axioms and the other possible axiomatization ... but I didn't want to make things too hard just from the start ... we need the time to get familiar with some concepts ... I'm gonna talk about that in the next videos ... thank you very much for you comment
@@guzmat Ok, thx. I've forgotten about all the elements C that are neither in A or B. Ooops! ;)
It is really weird that there are mathematicians do not believe in the existance of the creator !
We don’t need him (Him).
well. if you dont need him then you wouldnt exist actually @@pesselatchicomo9088
Mathematics has no single creator. It was created by countless individuals across millennia, each building on the works of their predecessors.
agreed!
Mate I get what you’re trying to say but I gotta slightly disagree with you. The axiom you name is this video is more of a definition of what it means for 2 sets to be equal. Also, the jar thing, we have no idea whether the elements of a jar are ordered or have specific coordinates, so the equality symbol there needs extra definition.
But I appreciate the attempt to teach rigorous logic to the world.
Yeah, I agree about the axiom being actually a definition ... but it is normally considered an axiom ... thanks for your comment ...
Interesting
awesome topic
Is this a footage of a ghost 😮behind GuzMat at 2:04?
My cat !!! "Pallino"
Ah, very nice 😘. By the way, thank you for covering axioms in mathematics. They are so widely unknown!!! I love the 1+1 = 2 question ❤️ almost everybody falsely thinks is an axiom. I disagree with your statement “you can trace the entirety of maths back to these nine axioms”. Think of what Gödel proved was never possible 🤓! Neither are these nine set-theoretical axioms the only ones in mathematics.
yes, you are right ... it is not entirely accurate ... it was just a way to introduce the subject ... we'll be talking about Godel Theorems later in the series ... thanks for you comment!!!
**ATTN: 😌😉This comment is equal to a rant from a novice mathematician who desperately wants and LONGS to know and understand mathematics.😔😉😁**
Question: if math and the basic foundations for math are so hard for people to comprehend-as your video has displayed-could it be the mathematical: i.e., theorems and axioms; system being taught is flawed???
Honestly, using logic, it is a known (“proven”) fact that a vast majority of people, globally, have issues trying to understand mathematics as they have been taught.
Therefore, could it be the commonly used Greek adapted mathematical, now called the Algebraic system, is grossly flawed???
How many more centuries must mankind endure this dated, and, in my opinion, flawed and grossly hard to comprehend Greek philosophical interpreted version of ancient (Egyptian and Babylonian) “earth measurements,” which is now commonly known as the globally accepted Grecian term, geometry (Merriam-Webster: Geometry- Etymology…. from Greek geōmetria, from geōmetrein to measure the earth)?🤔🧮 +🧐📚👨💻=😁🧑🏻🎓!
Hi, thanks for you comment.
First of all, are axioms really so hard to understand? I'm not sure, maybe some of them are.
But in any case the answer to your question is: yes, math could be actually flawed and this in itself is actually a proven theorem; not kidding.
@@guzmat, Aioxms might not be hard. However, as your video shows, math is not easily learned by many. Therefore, could it be the system is flawed? If so many people cannot digest, comprehend, and retain mathematics, we must realize reality. Certainly masses of people are not flawed. Instead, we must examine mathematics for there must be something flawed in the way mathematics are taught and interpreted. This is a global issue, so maybe other cultural forms of mathematics should be taught as an alternative option. Who knows, maybe there are more mathematicians in our world but they are not able to digest the Greek interpretation of mathematics.😉
Math can probably be taught better ... and it would be really interesting to discover different and new ways to do math. How do aliens do math? Is alien math the same as ours?
This whole business of axioms makes no sense to me as it relates to logic in mathematics. Consider, the statement 1+1=2 is almost tautological and true by definition. There is no reason for an axiom because the statement cannot be untrue. This is unequivocal. That the concept of 1 can be conceptualized in consequence of the perception of “a something by itself” and that label “1” arbitrarily formulated and applied to that concept means that that which was perceived is “1” something by itself. As long as all in witness agree that “1” means a something by itself, there is no need for proof. It cannot be otherwise. That another “1” or something by itself can be perceived and understood as such is by definition possible or the original something by itself could not have been perceived, which it was (and we named it).
By extension then, we cannot but understand that 1 and (or +) 1 is (1 and 1) or “2”, a second concept we formulate and name. The label 2 meaning a something by itself and another something by itself is thus by definition true. Again, no need for axioms or proof of its truth for were we unable to comprehend its unequivocal truth we could not have expressed the proposition to begin with which we did.
Again, by extension and the application of the same logic, we know that 2+2=4 by the same logic which we would be obliged to apply. The further extension of these simple propositions would be by the same logic, by definition true, unequivocally. What then is the point of these extensive, sometimes redundant axioms by which sophistry like that of Goedel’s claim of the “hole” in the logic of math can be imposed.
Any proposition which is true by definition requires no proof and so the insistence on the imposition of some analogously wordy and thus, unnecessary scheme is actually defiant of the logic it is purported to support or validate.
Any thoughts?
But is congruence identical to equality?
Well, it depends on what you mean by congruence ...
Equality is having the same elements ...
A congruence relation is any relation that is reflexive (a R a), symmetric (a R b implies b R a), and transitive (a R b, b R c implies a R c) ...
So in general congruence is more general than equality.
@@guzmat Equality describes the boundary, congruence fills to the boundary. The universe exists through its attempt to fulfill the Pythagorean relation, to find that integer where the hypotenuse is equal to B+3, although the relation is always congruent without integer satisfaction.
I think this video’s funny
1st is Axiom of Extension. Halmos, Naive Set Theory.
Way to long video for so little content. You could fit all axioms in one video instead inconvieniently dividing one simple topic on several videos.
thanks for writing the comment, some of the next axioms will be gropued in a single video ... but there are some that in my opinion need a video ... because the subject becomes complex very fast ... for example there are various ways to enunciate the second axiom and the equivalence of these different statements is really tricky. I think that some time is needed to familiarise with the concepts ... but I get what you mean and I'll try to be more concise.
Way too long? You can’t dedicate 7 minutes to one of the most important fundamentals of the known universe?
@@Samson484 I can dedicate it for better made material.
@@Dawid-kn6mv doubt it. Doubt you even understand the significance of this.
@@Samson484 I dont care what you doubt troll
👋👋
Tremaine Roads
1+1=2 is axiomatic per Peano's Axioms. the alleged proof which appears in Principia Mathematica is a circular application of this, and thus is unsound.
if 1+1=2 then there would be no need for algebra, and unit conversions would be impossible. consider, for instance:
1 foot + 1 yard = 48 inches
1 dog + 1 quail = 6 legs
1C water + 1C dirt = some mud
1 frog + 1 pond = 1 pond
you can thus soundly prove that a pair of vectors with comensurable units and magnitudes of 1 sum to one vector in the same unit with a magnitude of 2. but the number 1 cannot actually even be added to anything, and attempting to do so won't yield a number, let alone the number 2.
you can even see this in the orthography.
4-3 = 1; this is a subtraction, right?
-3+4=1; hang on, where did the + come from?
if what we're actually doing is using positive and negative vectors, like +4, -3, and +1, which have commensurable units, then it all makes sense. however, this also means that + and - are not operators, but partial names of the unit of these vectors. further, the grammar of the orthography we're using permits us to not write the + if it occurs in initial position within an expression. that doesn't mean it's not really there, it just means we're not explicitly saying it. such rules are quite common in human languages, actually.
now, reconsider the current foundation of mathematics armed with this understanding. nearly everything built up over the past two centuries assumes that + and - are mathematical operations which operate over bare numbers. but clearly this must be wrong. as such... what the hell are you doing?
I'm not sure I get everything you explain ... but thanks!
@mickyetanotherone3401 but what are you really trying to say? That 1+1=2 is an axiom?
1 + 1 = 2 because x + 0 = x, x + y = succ(x + pred(y)), pred(a) = b if succ(b) = a, succ(0) = 1 so pred(1) = 0, x + 1 = succ(x + pred(1)) = succ(x + 0) = succ(x), succ(1) = 2, therefore 1 + 1 = 2.
Of course, the fact that succ(0) = 1 and succ(1) = 2 are logically dubious, and defining them as anything else would still be completely consistent with math, but would break this "proof". Really most of this "proof" is highly dependent on the exact definitions of "0", "1", "2", "succ", and "pred", which is where the question of "what is an axiom" starts to become less about math and more about language.
Green Track
Well that is women for you
jump
Such a boring and repetitive narration. Entire video could be well condensed, and I cringed hard from the public polling
Binary code 0110111001(0-9) starts over again don't pass (10)
sorry I'm not sure I undestand what you mean ...
Okay look at your phone it goes from one to zero that's is all the numbers you need 1=abc 2=def all the way to 9=XYZ O stands alone (10) so if I was to SPELL something with numbers IGo by the alphabets or how equal they are spaced lets say 1to 9 =8 example 1+8=9 or 9-1=8
@@freddielittle9825 ok, i think I get it, thx ...