How the fuck do you define the natural numbers?
all non negative integers
>>7897915
dumb brainlet
Inductive nat : Set :=
| O : nat
| S : nat -> nat.
>>7897914
every whole integer greater than the amount of times that i have sucked on kristin stewart's toes
>>7897914
1, 2, 3, 4, ....
axiomtically
you dont define them, even a kid in primary school knows what are natural numbers
inb4 we can construct them through some set theory bullshit:
numbers are probably the most intuitive math concept, construction of natural numbers from sets is just an exercise in logic, but nothing else. Numbers are more obvious than set theory to most people, i doubt anyone ever thought
>wew i wasnt sure about those numbers, but now that ive seen von neumann construction of naturals i can accept them
set theorists just want to masturbate
>hurr look how good our field is, its the basis of all maths
while no one sane cares about their axioms and aleph milion bullshit
>>7897992
OK but that is kind of weird pham. Can you find a model of the Peano axioms without assuming the natural numbers.
>>7898011
This.
If someone unironically questions 1+1=2, then he has left math and went full philosotard.
My algebra prof once said that you can construct them with the help of the cardinality of the power set of the empty set. Let a be the empty set (= zero) , then the cardinality of the power set of a is 1, the cardinality of the power set of the power set of a is 2, etc. etc.
All numbers that occurs naturally in nature
>>7897914
Fuck you OP. Thanks to you I learned something today.
I was going through all the wikipedia pages for the natural numbers
https://en.wikipedia.org/wiki/1_%28number%29
https://en.wikipedia.org/wiki/2_%28number%29
https://en.wikipedia.org/wiki/3_%28number%29
...
https://en.wikipedia.org/wiki/999_%28number%29
and then I found
https://en.wikipedia.org/wiki/1000_%28number%29#Selected_numbers_in_the_thousands_.281001.E2.80.931999.29
>>7898017
>what is the finite field of two elements
>>7898085
>the cardinality of the power set of a is 1, the cardinality of the power set of the power set of a is 2
You want to recheck that anon.
Or get a new prof.
>>7898145
Oh, it's 2 and 3 right? Was not quoting my prof, just kind of outlining process
>>7898151
If |X| = n, then what is |P(X)|?
>>7898159
-1/12
>>7898143
>What is an isomorphism from that field to the field of rationales.
>>7897914
They appear in nature.
/thread
>>7898177
All of them?
>>7898016
There's only one (up to iso) in second order Peano artihmetic
>>7898085
For the (model of the) numbers themselves, you only need the empty set, {}, and
S(x) := x union {x}
Then let
0 := {}
1 := S(0) = S({}) = {} union {{}} = {0}
2 := S(1) = S{0} = {0} union {{0}} = {0,1}
3 := S(2) = S({0,1}) = {0,1} union {{0,1}} = {0,1,2}
and the n'th set has, accordingly, n elements.
https://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers
I'd also go with (>>7897925).
or maybe autistically implicitly characterized as an initial algebra, as in
https://en.wikipedia.org/wiki/Initial_algebra#Example
Are you too stupid to Google that exact question?
>>7897914
One, two,
Buckle my shoe;
Three, four,
Open the door;
Five, six,
Pick up sticks;
Seven, eight,
Lay them straight:
Nine, ten,
A big, fat hen;
Eleven, twelve,
Dig and delve;
Thirteen, fourteen,
Maids a-courting;
Fifteen, sixteen,
Maids in the kitchen;
Seventeen, eighteen,
Maids a-waiting
Nineteen, twenty,
My plate's empty.
>>7897914
I am going to get a bit technical here but:
1) Define a base element =1.
2) Define an operation that generates addition elements. Eg. Addition 1+1 =2.
3) Define All elements of the set of natural numbers as all numbers generated by the base element plus the operation.
So from this:
1 = 1
2 = 1+2
3 = 2+1
etc.
>>7897914
A natural number is defined as a transitive set, a set which always has an element that is an element of another element in the set except for 0 or whatever the starting natural number is.
set theory sucks dick as a mathematics foundation
unaesthetic af
>>7898386
Fuck you set theory is god-tier maths. Every field of maths should be taught using set theory.
>>7898420
set-theory is a pretty hideous formalism where everything is kind of arbitrary t b h.
I'll go the sperg way; a natural numbers object is the initial algebra of the endofunctor 1+(-) on a category C with terminal object 1 and coproducts.
In the category of sets, this is a set X with an element 1->X and a successor function X->X.
>>7898471
>set-theory is a pretty hideous formalism where everything is kind of arbitrary t b h.
Tbf everything can be formalised in many, or perhaps, infinitely many ways. There's no such formalisation that is not arbitrary.
>>7898168
This is bait, right?
>>7898471
So set theory is mental masturbtion, but this shit is not?
Category theory is autism++
Natural numbers refer to members of the set of positive integers 1, 2, 3, 4, etc...or the set of non-negative integers 0, 1, 2, 3, etc ... Still have disagreements whether to include 0 in the set.
>>7898530
Forgot to add, WHOLE numbers only. None with decimals, and no fractions.
A whole number must be able to represent a seperate and lone object in 3d space.
>>7898159
Not that anon, but clearly 2^n
>>7898521
It's incredibly useful, though.
>>7898168
>he thinks they are isomorphic
>>7898568
Oh god please be pretending to be retarded
A set with a lower bound and a function that sends members of the set to other members of the set with no cycles such that finite successive repetition of that operation leads to any element of the set
>>7898471
Category shills plz go, this is getting ridiculous
>>7897914
strokes on a blackboard
Set theory is NOT a good framework for defining the natural numbers. Is it like using nuclear weapons for pest control.
>>7899403
>Is it like using nuclear weapons for pest control.
>t.t*rk
nothing wrong with that
>>7897922
>chink monkeys
Typical brainlet-poster
>>7899403
Why are so many people that are butthurt about set theory?
>>7897914
n>0
n%1=1
Done
>>7899600
Because
it contains theorems that depend on the model and are meaningless and
it's axioms aren't actually used by any non-set theorists.
>>7899606
Assuming you can make sense of ">", the set {2,4,7} is already a model for the theory defined by that one axiom.
>>7899600
Because of the HoTT meme.
>>7898484
>There's no such formalisation that is not arbitrary.
>>7898594
It's correct, the sum of the combinations of k in n for k is equal to 2^n
>>7898011
>Numbers are more obvious than set theory to most people
The fact that time goes slower while getting closer to blackhole is more obvious than theory of relativity to most people.
The fact that you can use some theory to point things that can be understand without knowing it, doesn't mean that theory is bullshit.
>>7897914
definition =/= construction
It's an countable infinite set of objects with defined few operations on them.
It has total order, and we call the least element 0. The element that is larger than 0 and smaller than any other is called 1, etc. There is no greatest element in natural number set.
It also has defined +, -, *, /, mod, and many other operations.
>>7898192
this is the correct answer
>>7897914
by whatever thing you can invent so that the principle of induction holds. once you have plenty of such objects, you choose the smallest one or the easiest one to get.
>>7901521
>03
A few error. In naturals / and - are,t always defined, as + and * are. It's an ordered set of number you could use to count objects.
>>7901559
Right, it's not defined for every pair, but it is for some.
>>7900049
no u
intersection of all inductive sets
t. first year course in analysis
>>7897914
How the fuck do I get a girlfriend that looks like that?
>>7897925
why is type theory so perfect bros
>>7902748
in eating her pussy
0 = {}
1 = {0}
2 = {0,1}
3 = (0,1,2}
...
n+1 = {0,1,...,n}
Not all that hard.
>>7903139
>>
>Not all that hard.
it took 5000 of civilization to get this.
>>7897914
a set of numbers where any two given numbers' product is a part of that set.
>>7904197
>branch of mathematics that created many tools useful in different branches of mathematics and was developed by many great mathematicians
>no one gives a shit about it
What?
>>7904192
retard
