/sci/ - Science & Math

>>9242645
>What's your favorite axiomatization of the real numbers?
>The usual way to do it seems to be to view them as a totally ordered field that is Dedekin-Complete, or as Cauchy Sequences of rational numbers.
That's not an axiomatization, that's a construction of the real numbers.
Axiomatization is what you wrote below.

>>9242653
What this anon said, more or less.

You wrote a set of axioms. Also, since you didn't give axioms for multiplication then you don't necessarily have a multiplication in your models (and if a multiplication is given then there's no requirement that it resemble the one we're used to).

The 14 different axioms are given such that any model of the reals is isomorphic to any other. That is to say that we can reliably interpret them as the real numbers, this is not true of your 8 axioms.

Regarding constructions:
>These refer to the way in which one can encode a model of the "real" numbers into another axiomatic system such that they satisfy the axioms.
>There are two standard approaches for set theory. Both involve encoding "integers" and "rationals" as sets in the same way (with "addition"s and "multiplication"s encoded in set theory as well).
>The first approach takes the "rationals" and defines dedekind cuts over the rationals. After defining multiplication and addition between cuts one defines the set of all cuts and shows that it satisfies the axioms for the reals.
>The second approach is similar except your real numbers are defined as cauchy sequences (which are again defined as sets).
Whenever trying to encode the real numbers into another theory one must note that only a countable subset of the reals can be explicitly described in the theory and so the approach must rely on the remaining reals "existing" only in the models of the theory and not in the theory itself (in general it's only possible to explicitly describe at most a countable number of things, regardless of theory or language).

>>9242692
>Also, since you didn't give axioms for multiplication then you don't necessarily have a multiplication in your models.

Wouldn't that be a positive? An axiomatization should describe the structure with as few rules as possible. If we can do it without introducing new symbols, isn't that just good and we can add definition of multiplication later if we need it. The standard way of constructing from a ordered field also doesn't have derivation, integrals or tan operation and nobody complains.

>>9242797
>Wouldn't that be a positive?
Depends. I'll give an informal example. Suppose I tell you I'm going to give an axiomatic system for squares and I do it by giving the following axioms (ignore that one of these is redundant, it's not the point I'm trying to make):
>All squares have four sides of positive length.
>Opposite sides are the same length.
>Adjacent sides are the same length.

Now suppose someone else comes along and says they can give a better set of axioms for Squares like so:
>All squares have four sides of positive length.
>Opposite sides are the same length.

Now it's true that their system is shorter but it doesn't just describe squares. In fact rectangles also satisfy these axioms. In other words, it is a perfectly fine axiomatic system, but it is an axiomatic system for rectangles, not squares. Though I could always use the rectangle system and define a square as "a rectangle satisfying some extra properties" but there should usually be some reason for doing so. I mean, there's nothing really stopping you from replacing lots of axioms with definitions.

>An axiomatization should describe the structure with as few rules as possible.
Not necessarily true. Your set of axioms should contain independent axioms (no axiom should be derivable from the rest), that much is true. However, when choosing axioms you should choose ones that capture some intuitive concept and are easier to work with. You're not doing yourself any favors by choosing an axiomatization that is hard to use and remember just because there's less axioms (especially since the axiomatizations ought to be equivalent).

>If we can....
>The standard....
Do you think multiplication is sophisticated addition? Try defining [math](\pi)(e)[/math] in terms of addition.

>>9242797
>Wouldn't that be a positive? An axiomatization should describe the structure with as few rules as possible
But if you don't define multiplication then you haven't defined the structure of the reals as a field. There is no one algebraic structure of the real numbers, rather there are multiple objects in different categories obtained by adding or removing structure from or to the real numbers in some other category.

>>9242830
Actually those are axioms for quadrilaterals :)

>>9242830
not that anon but does this mean you have to write enough of a description to include all the properties of the real numbers, and not just enough to construct them, to be considered THE axiomatization of the real numbers?

>>9242645
The one where they don't exist, because they really don't.

>>9242945
>maths cares about metaphysical questions like whether concepts like 'existence'
really adds the wild onto the 'berger

Not Cauchy Sequences, so much is clear.

>>9242892
Fuck, you're right! I will go commit sudoku now.

>>9242908
Not exactly. Really it all comes down to what you mean when you say "real numbers".

Here's an excerpt from Bertrand Russel's Introduction to Mathematical Philosophy:
https://people.umass.edu/klement/imp/imp.html (ctrl+f "[page 27]")
>We now know that all such views are mistaken, and that mathematical induction is a definition, not a principle. There are some numbers to which it can be applied, and there are others (as we shall see in Chapter VIII.) to which it cannot be applied. We define the “natural numbers” as those to which proofs by mathematical induction can be applied, i.e. as those that possess all inductive properties. It follows that such proofs can be applied to the natural numbers, not in virtue of any mysterious intuition or axiom or principle, but as a purely verbal proposition. If “quadrupeds” are defined as animals having four legs, it will follow that animals that have four legs are quadrupeds; and the case of numbers that obey mathematical induction is exactly similar.
>We shall use the phrase “inductive numbers” to mean the same set as we have hitherto spoken of as the “natural numbers.” The phrase “inductive numbers” is preferable as affording a reminder that the definition of this set of numbers is obtained from mathematical induction.

This is a pretty oldschool reference and not everyone agrees with him but I think he hits the nail on the head here. To him the "natural numbers" just means a structure that supports induction. That is to say that the "natural numbers" only need a starting "number", and a way to reference the next "number" at each "number"; but these numbers aren't necessarily numbers in the way we usually think of them (eg. there isn't a 3 that represents 3 objects but there is a S(S(1)) that represents the third number). Counting numbers, inductive numbers, natural numbers, etc.. could all mean different things to different people.

>>9244604
But induction can be used on any well ordered set, and since any set can be well ordered we can use induction on any set, like reals.

>>9245113
So there are variants of induction that function on the reals (and some for other topologies as well) but for the most part when people talk about induction on the reals they're really talking about induction on the naturals where the naturals they're referring to are just some countable subset of the reals that act as a model of the naturals.

>>9245223
Principle of induction, the one we're familiar with, not some transfinite variants, applies also to {1,1.5,2,2.5,...}

>>9242645
The biggest archimedian abelian totally ordered group

>>9245431
Yes. Note that {1,1.5,2,2.5,...} has a starting element and each element has a next element. So it can be interpreted as a model of the naturals.

I think you're stuck on the idea that the naturals are numbers like 1, 2, 3... and not grasping the idea that natural numbers don't really refer to "numbers" in any usual sense.

I am enjoying this thread and would like to thank all participants

>>9245450
I mean, this set, but where 1.5 is not a successor of any other number, so is has two starting points, so it's different object than naturals and yet principle of induction is valid on this set

>>9245500
1.5 is the successor of 1 in that set. That is how you're doing induction on it, is it not?

>>9245509
I think he means the union of two "different" natural numbers.

>>9245532
I'm not sure if I'm misunderstanding here. There is no problem with doing that and performing separate inductions in parallel. Similarly there's no problem with taking some set and performing inductions on it in different ways (instantiating it or subsets of it as models of the natural numbers in different ways to do so).

Ultimately what makes something the naturals is the way that you do induction on them, just as in Russels description.
>We define the “natural numbers” as those to which proofs by mathematical induction can be applied, i.e. as those that possess all inductive properties.
>If “quadrupeds” are defined as animals having four legs, it will follow that animals that have four legs are quadrupeds; and the case of numbers that obey mathematical induction is exactly similar.
In other words, if natural numbers are defined as a set upon which you can do induction then it follows that sets upon which you can do induction are natural numbers.

>>9245440
So hold on, we can show the existence of the real numbers nonconstructively using Zorn's lemma?

>>9245590
Nonconstructive existence is an oxymoron.

>>9245665
Maybe in ZF :^)

>>9245812
The way choicefags use double negation is literally retarded. Imagine that sort of reasoning being applied to the real world:
>Suppose reptilians don't exist.
>That conflicts with my worldview where reptilians run the government.
>Therefore it must not be the case that reptilians do not exist.
>Reptilians exist, QED.

>>9245831
>the cartesian product of nonempty sets may be empty
your average intuitionistfag everyone

>>9245847
In order to prove that the infinite cartesian product is non-empty you need to demonstrate an element inside of it. So while you may be able to give a finite description of an element in special cases the actual task becomes impossible in general.

Sooo, choicefags never actually do it. They just suppose things about supposed elements inside of it.

>>9245861
just pick an element from each set :^)

>>9245665
>Krull's lemma shows every field has an algebraically closed extension
>reeeeeeeeee that's nonconstructive!!!!
How can you not flunk undergrad math completely with that autistic screeching?

>>9245911
constructivebros:
>Thing is true in cases we care about and the proofs for those cases let us demonstrate things explicitly.

non-constructivelets
>reeeeeee thing is true for cases we don't care about even if the proofs for it are useless to all.

lmao, keep dreaming brainlet.

>>9245900
zoz

>>9245934
>he doesn't know how useful nonconstructive existence proofs may be
confirmed for not doing any order theory, real analysis, graph theory or even fucking calculus
go fap to your cs-tier meme math some more

>>9245113
>any set can be well ordered

>>9245590
>existence
>nonconstructively
I think you mean "non-refutation of existence" there.

>>9245847
>the cartesian product of nonempty sets may be empty
Literal retard and or freshman detected, kill yourself. I wouldn't be surprised if you think that because excluded middle isn't provable constructively, there are somehow propositions in constructive foundations which are neither true, nor false.

>>9246314
>order theory, real analysis, calculus
All of those are engineer-tier garbage. They should be banned and outlawed.

foundations dart game

>assume R exists
>state a bunch of "intuitive" facts about the reals
>see how much of ZFC you can derive from your axioms

hard mode: you cant pretend that the existence of a well-ordering of the reals is intuitive
you may assume that the category of categories has equalizers

>>9246674
>assume R exists
It then trivially follows that [math]\mathbf{Cat}[/math] has a zero object, which is a contradiction.

>>9245545
The set {1,1.5,...} where 1.5 is not a successor of any other number isn't the same as naturals because the two objects have different properties, and principle of induction applies to this set, so principle of induction is not enough to define naturals uniquely, but it's true that naturals are the smallest inductive set and this property determines them uniquely up to isomorphism

>>9245509
No, 2 is successor of 1, 2.5 is successor of 1.5 and 1.5 is not a successor of any other number. My point is you can construct sets strictly larger than naturals (larger regarding inclusion, not necessarily cardinality) that also follow the principle of induction showing that it's not enough to describe naturals uniquely

>>9246684
thats actually not true

>>9246616
What's your problem? That's equivalent to the axiom of choice

>>9246748
>axiom of choice

>>9246748
The axiom of choice is false though.

>>9246741
>>9246736
>thinking successor means +1
lmao, I see now why you're so deadset on being wrong.

no, anon. If you're doing induction on a set then you have a starting point and at each point you have a next.

If you're saying that 1.5 is not the successor of 1 then you can't do induction on it.

>different properties
Those aren't properties of the natural numbers.

>>9245113
Well ordering talks about subsets of reals, not the reals themselves. In order to give the reals a structure akin to the natural numbers you basically need to prove they're countable and your induction has to follow the order given in your list.

The rationals are countable and they can be listed and in that sense can be perceived as a model of the natural numbers. However, to do induction on them using that principle then your induction must follow that order.

>>9246630
This.

>>9246314
>he doesn't know that all of those "useful nonconstructive existence proofs" (my sides) can be replaced by constructive existence proofs on a statement that's equivalent under classical logic but not equivalent under intuitionistic logic. So basically people doing nonconstructive proofs are just doing inferior work out of ignorance.

>>9246837
But subset of reals are neither countable and you'll find always bijections from an interval to IR.
So with AC it is possible to enumerate the reals. The witchcraft is transfinite induction.

>>9246855
Transfinite induction is not that fancy nor does it rely on an enumeration of the reals (just an ordering).

An enumeration is a list.
An ordering is a statement about all subsets having a least element.

Really transfinite induction is just a pretentious version of the real induction described on page 5 here:
http://math.uga.edu/~pete/pointset.pdf
and that's just a special case of induction on sets with the order topology. There's nothing witchcraft about it.

>>9246902
Possible that I misunderstood something. But:
In the proof of Zermelo's Well-Ordering Theorem transfinite induction is explicitly used
to enumerate an arbitrary set.
I mean it makes sense: If you can well-order a set, then you should be able to write it
as a list (enumerate it).

>>9246919
>If you can well-order a set, then you should be able to write it as a list (enumerate it).
>enumerate it
So every well-ordered set is countable, I see.

>>9246924
No. But how do you well order a set (i.e. proof Zermelo)? You use a choice function to create a
transfinite sequence that _enumerates_ the set. I mean how would you do it in a different way?
On the other hand, if A is well-ordered. Take the least element a. Well order A-{a} and take the least element. Repeat this and you will get a list [math]a_1,a_2,...,a_{\omega_0},...a_{\omega_1},...[/math].
As I said, possible that I misunderstood something, but then explain it instead of making useless posts.

I understand defining a total ordering as a binary relation ~ that is anti-symmetric, transitive, and total.

Intuitively, we could construct the relation ~(1,2) , ~(2,3) , ... but this is arbitrary because the definition is well defined for ~(1,8) , ~(8,3) , ... or any other order.

My question being, is there a way to inject the ordering 1, 2, 3, 4, ... without relying on arbitrary constructions, intuition, or more axioms?

Can someone explain to a brainlet why cauchy sequences construct the real numbers? How do I KNOW there isn't some real number that a cauchy sequence can't approach? Can I construct the reals with sequences of irrational numbers?

>>9247022
>construct the real numbers
Impossible.

>>9246774
>chosing one element from each set is false

>>9246834
>Those aren't properties of natural number
Of course they aren't, that's the point, there exist objects different than naturals on which we can use induction

>>9246852
Constructive proofs are for brainlets and engineers (who are brainlets too)

>>9247112
So the vast majority of mathematics is for brainlets?

>>9247124
Yes

>>9247022
The real numbers are defined as precisely the numbers that you can approximate arbitrarily well using the rationals, or the closure of the set of rationals. Just think in terms of decimal expansions, which are Cauchy sequences of rationals that all have a limit in R. The irrationals would also "construct" R, as they are also dense in R, but it wouldn't have any point since you have to know what the real numbers are to know what the irrational numbers are.

>>9242653
That's actually an axiomatization.

>>9247141
Spotted the subhuman engineer.

>dude I can't use this object if I don't know how I can construct it
>not exactly a brainlet engayneer mindset

>>9247214
>object
This already implies constructive existence (there is no other kind).

>>9247206
Spotted the "if I don't see explicit construction of an object then it doesn't exist" brainlet

[math]4+4=74[/math]
[math]4+4=74[\math]

>>9247112
You have that backwards, son. The people who rely on choice only do so because they want the theories to act like "muh real world". Meanwhile ZF people are perfectly happy with having 8 different and incompatible definitions for infinite sets because to them the notion of a set doesn't mean "lol a collection like in muh real world".

So no, non-constructive proofs are for engineers, scienceplebs, and other sorts of brainlets.

>>9247225
>if I can't observe it it doesn't exist

>>9247646
Yeah, knowing that every surjection in Set splits is necessary only for engayneers caring about muh real world

>>9247646
>infinite sets

>>9247646
>a function is injective iff it has a left inverse, which is a surjection
>a function is surjective iff it has a right inverse, which is an injection
takes a special kind of retard to accept the first and reject the other

>>9247107
You're not paying attention.

The natural numbers are not necessarily numbers. The only properties of the natural numbers are
>it has a starting point
>each point has a next point called a successor
and therefore one can do induction on them. Basically any infinite list is an example of the natural numbers regardless of whatever the list holds. The induction is performed based on the enumeration given in that list.

So, if I give an enumeration of the "rationals" then that enumeration is the natural numbers and I can perform induction on them.

All of these other properties you're thinking of have nothing to do with induction and aren't really properties of the naturals so much as they are properties of the counting numbers or of the non-negative/positive integers.

>>9246924
Thank you.

>>9246919
>>9246937
You did misunderstand. Enumeration is not an ordering and no an order does not give you an enumeration. This ought to be obvious because as the other anon pointed out, an enumeration would imply that the set is countable.

>>9247022
The problem isn't that there is a real number that a cauchy sequence can't approach. The problem is that it is impossible to give a finite definition (or description) of the cauchy sequence describing a real number in almost all cases.

In particular, the set of real numbers for which it is possible to produce a finite definition (or description) is countable. The set of real numbers for which it is impossible to produce a finite definition (or description) is uncountable.

That means that for the vast majority of these reals you actually have to write out the infinite sequence explicitly which means statements about those numbers are unprovable and it's basically a garbage construction.

>>9247677
And where did I say the naturals are necessarily numbers? Here, my example without numbers which cause your tiny engineer brain to overheat, {a,A,b,B,c,C,...} where every lowercase symbol different than a is successor of other lowercase symbol, each uppercase symbol different than A is a successor of other uppercase symbol, no uppercase symbol is a successor of lowercase symbol and no lowercase symbol is successor of uppercase symbol. Set with subset isomorphic to naturals, a successor function and two (and not just one like in N) starting points.
I know merely relabeling elements of N doesn't change it's fundamental properties, I'm not some fucking braindead idiot like you, but that's not what I've done

>>9247662
>>9247676
1) The terminology you're looking for are
>split epimorphism
>section
>retraction

2) Axiom of choice is only needed for retarded sets.

3) If you're working in the category Set then you're better off defining the category synthetically and defining your problem sets with limits (which are part of your definition of the category). In other words, you are 1000% a brainlet if you're tripping yourself up over the technicalities of axiomatic set theory while working in category theory.

>>9247677
>Basically any infinite list is an example of the natural numbers regardless of whatever the list holds.
no, the naturals don't have any limit ordinals

>>9247693
lol, explain how you do induction on that set and I'll tell you how you are retarded.

>>9247715
>induring retard who doesn't know what an enumeration is

>>9247719
>induring

>>9247724
What is it like to be cancer?

>>9247716
That's trivial

>>9247736
I'm glad we agree that you are trivially retarded.

>>9247716
Assume some statement is true for all m less than or equal to some M and show it's true for succ(m) for all m≤M

>>9247760
>Two separate inductions on lists [a,b,c,...] and [A,B,C,...]
>lel their union is the naturals because I did induction on them

>>9247719
You cannot enumerate an uncountable set, even if it's well ordered. Every element having a successor doesn't mean it will also have a predecessor.

>>9247775
I never implied otherwise. I suggest you re-read my post. In case it's not clear, by list I mean enumeration.

>>9247766
For fuck's sake are you retarded or something? That object is not naturals, and that's my point. What I'm trying to say is that the principle of induction is not enough to uniquely (up to isomorphism) define naturals

>>9247783
Enumeration is setting bijective correspondence between elements of the set and subset (proper or not) of naturals, therefore any set with cardinality greater than cardinality of naturals can't be enumerated

>>9247899
Again, check your reading comprehension anon. I am not the same brainlet who thinks you can enumerate the reals via well ordering. That anon is a fucking retard.

>>9247894
The greentext was me quoting you. That's how you sound, halfwit.

>>9247908
Then how
>This object is naturals
is quoting me if I said
>This object is not naturals

>>9247915
Then how
>I can perform induction on this object
is quoting me if I said
>I can perform induction on this object

fixed, and no you aren't performing induction on that object.

>>9247919
>I'm an idiot who thinks induction can be performed only on naturals therefore everybody who is not braindead is wrong

>>9247926
>We define the “natural numbers” as those to which proofs by mathematical induction can be applied, i.e. as those that possess all inductive properties. It follows that such proofs can be applied to the natural numbers, not in virtue of any mysterious intuition or axiom or principle, but as a purely verbal proposition. If “quadrupeds” are defined as animals having four legs, it will follow that animals that have four legs are quadrupeds; and the case of numbers that obey mathematical induction is exactly similar.
-Bertrand Russel

Induction is an axiom of the naturals. It isn't an axiom of "lel look at this arbitrary set I just made" so in order to apply it we instantiate the set as a model of the naturals.

>>9247939
For fuck's sake, maths is not about what somebody smart and important said, but about what's true. Just grab any textbook and if it will list induction as one of defining properties of naturals then it will also give couple others because, as I've already said, principle of induction doesn't define naturals uniquely

>>9247201
So how would you call the difference between defining real numbers by their field properties, etc... and explicitly constructing them via Dedekind cuts or Cauchy sequences?

>>9247971
>explicitly constructing them via Dedekind cuts or Cauchy sequences
This is called math.
>defining real numbers by their field properties, etc
This is called bullshit.
Just giving a list of properties doesn't prove that there actually exists an algebraic structure which satisfies all of them.

>>9247971
Axiomatization vs "Construction" (not to be confused to constructive).
>>9247979
You'd still need to axiomatize the Natural Numbers and set theory. Any other foundation for mathematics will require this too if you wish to retain the logical rigor that we have come to appreciate. (Need axioms in order to prove theorems)

>>9247979
>Just giving a list of properties doesn't prove that there actually exists an algebraic structure which satisfies all of them.
Yes, it does since these properties are the extrapolation of what we want geometrically from reals.
In the same fashion we derive properties for natural numbers.

>>9247899
>>9247904
Enumerate is about ordinals not cardinals.
And you can use "enumerate" with limit ordinals ("enumerate an uncountable set via transfinite sequence").

>>9247961
>then it will also give couple others because, as I've already said, principle of induction doesn't define naturals uniquely
https://en.wikipedia.org/wiki/Peano_axioms#Formulation
Sounds like you're confusing the naturals with something else then, anon.

Protip: arithmetic aren't part of the axioms of the naturals, they're just definitions one can construct on top of them.
Protip: S(n) = n+1 is NOT part of the axioms of the naturals. That is just part of the arithmetic used in common conventions (in fact "1" isn't even a thing in the naturals and "0" only refers to a starting point, not 0 with respect to algebraic operations like in a ring.

>>9247971
>axioms
This is the actual axiomatic system for the reals.

>explicitly constructing them via Dedekind cuts or Cauchy sequences
This is done in axiomatic set theory, but the idea is that you are just creating data structures and rules that ultimately satisfy the axiomatic system for the reals.

>>9247979
This person is a brainlet who doesn't know dick about formal logic.

>>9247979
>t. elitist freshman.

>>9248005
>using different definition of enumerable when anons are explicitly talking about lists.

pls respond
>>9246945

>>9248021
>>9246945
Your post is too unclear. I have no idea what you're asking. Is ~(a,b) your way of saying a~b?

>Let S be the set of all totally ordered field extensions of Q.
>As Q is an element of S, S is nonempty.
>S is partially ordered under set inclusion, and every chain in S has an upper bound, namely the union of the chain (proof: think).
>Therefore by Zorn's lemma, S has a maximal element, say R.
>Assume there exists an upper bounded subset T of R with no least upper bound
>Construct a field extension R(t) with ordering induced by the ordering on R such that t ≤ any upper bound of T and ≥ any other element
>By assumption, this is a field extension of Q that is larger than R, contradiction
>Therefore R is Dedekind complete
should piss off both constructivelets and nonconstructivelets

>>9247939
>>9247961
Are you guys aware that this Bertrand Russell quote was written like 100 years ago ? Math conventions, vocabulary and notations change as time passes. What we call induction nowadays isn't what Russell was referring to.

Yes, we can do induction on other structures as N, like transfinite induction or more generally well-founded induction (see https://en.m.wikipedia.org/wiki/Well-founded_relation for details). The "two-lists" induction thing is one of them.

But that's completely irrelevant to that quote, since Russell is obviously just talking about the induction on N, with one starting point and a successor function. This definition of N as "any structure which supports this induction principle" is what we nowadays call a "natural number object" in category theory.

>>9248030
I didn't invent the ~(a,b) notation, it is the notation used for binary relations in axiomatic set theory.

You can define a~b as materially equivalent to ~(a,b) for rotational convenience but I don't see the need for it here.

>>9248055
Stop being autistic you're communicating with actual human beings. a~b is more readable than ~(a,b) no matter the context, so write it like. Also since it's an order relation write it a < b, even better.

>>9246945
The ordering 0 < 1 < 2 < 3 < ... on natural numbers is a total order.
But not every total order looks like this.
Does that answer your question ?

>>9248071
I disagree with your post and have no interest in an argument. I will not reply further on this point.

>>9248076
I understand what a total order is. The possible total orders of IN is infinite and as far as I'm aware the conventional ordering is arbitrarily chosen.

Again, my question is can we construct the conventional ordering without making an arbitrary choice? Can we construct the conventional ordering without arbitrary choice or additional axioms?

>>9248095
>without additional axioms
If you don't add additional axioms then what you have is a total order, and you said yourself that not all of them look like (N, <).
What can you do besides adding axioms ?

>Let < be a total order
>Not allowed to state any other properties of <
>Somehow < should become the ordering on natural numbers

>>9248007
For fuck's sake, I've just explicitly constructed an example showing the induction doesn't uniquely define naturals, what's more to say about that

>>9242645
1. R is a field
2. R is ordered
3. R is complete

QED.

>>9248148
Are you just pretending to not understand ? What they're calling induction is "forall P, P(zero) and (forall x, P(x) => P(succ(x))) => forall x, P(x)".
Your counter-example doesn't satisfy this principle.

>>9248147
I'm sorry I couldn't convey my question any clearer, but you do not understand. Further discussion would be inappropriate.

>>9248148
Your set is inductive.
>{m,M,n,N,...}
But when performing induction with this set, as you defined the set, you would typically split the set into two subsets and pick one.
>{m,n,...}
>{M,N,...}

>>9248177
So what, what's your point?

>>9242645
Separation is false...
Let's say Y = { y | y >= 0 } and X = { x | x < 0 }
For every x, y, x < y. But there is no z that respects x < z < y

>>9247245
>>9247659
Subhuman engineers are so fucking retarded. Why are non-mathematicians even discussing this?
Refutation is not the same as the inability to prove something. Please at least try to understand that.

>>9248055
That's wrong though.

~(a)=b is function notation meaning ~ evaluated at a is identical to b.

a~b is relation notation meaning the the tuple (a,b) is in the set ~.

~(a,b) is not a standard notational convention.

>>9248052
That is both what we call a natural number object and what the naturals actually are as an axiomatic system in general mathematics. Moreover, induction has to be defined at the axiomatic level because we are actually creating a universal quantifier. It is not self evident and isn't something you can prove. You may have convinced yourself otherwise because induction is intuitive but you are wrong. All of those other inductions are DIFFERENT THINGS.

Learn some basic mathematical foundations or gtfo.

>>9248148
Kill yourself, retard.

>>9248314
I'm not those anons but basically you're not talking about induction. You're talking about bullshit and spazzing out while others try to explain how you're wrong.

>>9249000
I'm afraid not. In fact, usually the notation of a binary relation is the ordered triplet (a,b,G) where G is a subset of the Cartesian product AxB.

However, when ~ is the characteristic function of G such that:

~: AxB ⟶ {0,1} | (a,b) ∈ G ⟼ 1 , (a,b) ∉ G ⟼ 0

Then ~(a,b) is the standard notation. Unfortunately, this will be my final post in the thread.

>>9249000
>~(a,b) is not a standard notational convention.
You're wrong, it is standard and it's called prefix/polish notation.
Obviously you never looked into a set theory or logic book.

>>9249120
Not the same thing >>9249111 is talking about.

>>9249111
Okay, I looked it up. Apparently that characteristic function notation is standard, though I'd personally never seen it written that way.

>ordered triplet (A,B,G) where G is a subset of the Cartesian product AxB
fixed

The non-triplet definition is also used and means something different (has no domain/codomain, and with the special case of functions it refers to a class of functions distinct from those described in the category set).