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/sci/ - Science & Math

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11209909 No.11209909 [Reply] [Original]

How do we know that the two statements
>There is no largest natural number
>The natural numbers go on forever
are equivalent? Never mind whether this implies that the set is infinite or not, how can we prove that it isn't just a finite poset?

>> No.11209936

>>11209909
What are the two new lights in the middle?

>> No.11209940

>>11209909
"Go on forever" isn't exactly a mathematical term. What do you want that to mean?

>> No.11209944

>>11209936
Military/air bases I assume.

>> No.11209945

>>11209940
For all x in N, x+1 exists

>> No.11209956

>>11209945
Consider ℕ \ {10}

>> No.11209974 [DELETED] 

>>11209945
Then they aren't equivalent. Not having a largest element isn't enough information to distinguish N as a totally ordered set. R doesn't have a largest element either.

The property that x+1 exists for all x in N is part of the Peano axioms. It's built into the definition of the naturals.

>> No.11210041

>>11209956
(Not op) what does [math]\mathbb{N}[/math]\[math]{10}[/math] means?

>>11209909
The natural numbers are uncountable. This is clear from their construction using Peano's axioms.

>> No.11210065

>>11210041
>The natural numbers are uncountable
nope, but real numbers are

>> No.11210067

>>11210041
>(Not op) what does [math]\mathbb{N}[/math]\[math]{10}[/math] means?
The natural numbers, except for 10. Then 9 doesn't have a successor.

>> No.11210070

>>11210067
>Then 9 doesn't have a successor
why isn't it 11?

>> No.11210085

>>11210070
>why isn't it 11?
because 9 + 1 is not 11

>> No.11210088

>>11209945
Even if x is the largest natural number, then x+1 still exists, it just isn't a natural number.

>> No.11210095

>>11209909
The natural way of putting your two > into formal terms would be the same, namely that for all natural numbers, there's a new bigger natural number. The prove is to map x to x+1.
The assumption in that is that N is well ordered, which is the case due to their axiomatization. This holds for all the models.

>>11209956
The set given by "N without 10" just isn't the set N we talk about here in the two > statements

>> No.11210117

>>11210095
>The set given by "N without 10" just isn't the set N we talk about here in the two > statements
Sure, but when you fix that set then the two statements are trivially equivalent because they're both true, right? That would make the question uninteresting.

>> No.11210119

>>11210088
>>>/x/

>> No.11210507

>>11209909
Tao's "analysis 1" Chapter 2 starts from this and explains it quite clearly. Download it from li bgen. Don't listen to the meme answers in here.