Is 1>0?

[thispersonisagenius]

The most basic of foundations for mathematics stems from the fact that 1>0, 2>1, etc. But is this true? What if the essence of 0 is greater than the essence of 1, but our stupid human minds interpreted it in the reverse? I asked this question in chat, and someone said that "it was not postulated [that 1>0], but set". What if, in a parallel universe, 0>1? How would everything work?

Please post your thoughts below, as this actually is a legitimate question.

[Cheesy111]

Nope.  0 and 1 are representative of amounts.  If 0 > 1, let's say that 0 is 1 and 1 is 0, then everything would be the same just called by different names.

It's like any word, really.  The word "word" might not mean the same thing in another language or universe, but that doesn't matter.  It's used to communicate an idea, no matter what word is used to do so.

[choongmyoung]

Speech is Silver, Silence is Golden.

[mesaprotector]

As a mathematical logic student, I beg to differ.

Let's consider a universe in which (ℕ,+,x) is defined exactly in our normal system. (Addition and multiplication do not change.) However, the ordering function < is inverted, as likewise is the successor function S (such that S(x) = x-1 instead of x+1).

This would not affect the familiar axioms of commutativity, associativity, and distributivity. However, when we get to the Peano axioms (most of which use the successor function), our world is turned upside down.
Let's look at them in their original forms:

1. 0 is a natural number. (This does not change.)
2. The successor of a natural number is another natural number. (Unfortuately, we have to scrap this, since the successor of 0 is -1, which is not a natural number.)
3. 0 is not the successor of any number. (Not true, it is the successor of 1.)
4. Different numbers have different successors. (Still true.)
5. The induction principle - if you have a statement about numbers Th, such that Th(0) is true, and Th(n-1) implies Th(n) for any natural number n, then that statement is true for all natural numbers n. (This does not work at all, and unlike statements 2 and 3 above, which can be revised so they still have meaning, there is no good way to fix the problem here.)

Without induction, many previously simple proofs become very difficult and/or impossible. Number theory becomes the ability to play around with calculators, and many other branches of mathematics become obsolete.

[choongmyoung]

As a mathematical logic student, I beg to differ.

Let's consider a universe in which (ℕ,+,x) is defined exactly in our normal system. (Addition and multiplication do not change.) However, the ordering function < is inverted, as likewise is the successor function S (such that S(x) = x-1 instead of x+1).

This would not affect the familiar axioms of commutativity, associativity, and distributivity. However, when we get to the Peano axioms (most of which use the successor function), our world is turned upside down.
Let's look at them in their original forms:

1. 0 is a natural number. (This does not change.)
2. The successor of a natural number is another natural number. (Unfortuately, we have to scrap this, since the successor of 0 is -1, which is not a natural number.)
3. 0 is not the successor of any number. (Not true, it is the successor of 1.)
4. Different numbers have different successors. (Still true.)
5. The induction principle - if you have a statement about numbers Th, such that Th(0) is true, and Th(n-1) implies Th(n) for any natural number n, then that statement is true for all natural numbers n. (This does not work at all, and unlike statements 2 and 3 above, which can be revised so they still have meaning, there is no good way to fix the problem here.)

Without induction, many previously simple proofs become very difficult and/or impossible. Number theory becomes the ability to play around with calculators, and many other branches of mathematics become obsolete.

Actually it starts with 1, not 0. Then 2 is defined as the next number of 1. 3 is the next.
Then, we define - as the inverse function of +. 0 is defined as x-x.

[eaglgenes101]

As a mathematical logic student, I beg to differ.

Let's consider a universe in which (ℕ,+,x) is defined exactly in our normal system. (Addition and multiplication do not change.) However, the ordering function < is inverted, as likewise is the successor function S (such that S(x) = x-1 instead of x+1).

This would not affect the familiar axioms of commutativity, associativity, and distributivity. However, when we get to the Peano axioms (most of which use the successor function), our world is turned upside down.
Let's look at them in their original forms:

1. 0 is a natural number. (This does not change.)
2. The successor of a natural number is another natural number. (Unfortuately, we have to scrap this, since the successor of 0 is -1, which is not a natural number.)
3. 0 is not the successor of any number. (Not true, it is the successor of 1.)
4. Different numbers have different successors. (Still true.)
5. The induction principle - if you have a statement about numbers Th, such that Th(0) is true, and Th(n-1) implies Th(n) for any natural number n, then that statement is true for all natural numbers n. (This does not work at all, and unlike statements 2 and 3 above, which can be revised so they still have meaning, there is no good way to fix the problem here.)

Without induction, many previously simple proofs become very difficult and/or impossible. Number theory becomes the ability to play around with calculators, and many other branches of mathematics become obsolete.

Or we could flip mathematical definitions that depend on pos/neg.

[cometbah]

We begin with an empty set, represented as:

{}

We construct a set that contains the empty set:

{ {} }

We construct a set that contains both the set that contains the empty set, and the empty set itself:

{ {}, { {} } }

We construct a set that contains the previous set, the one before the previous set, and the empty set:

{ {}, { {} }, { {}, { {} } } }

And so on.

Now, we define something called 'cardinality', and put each constructed set into a 1-to-1 corresponding relationship with each 'thing' that is a cardinality:

{}                                   -----------> Thing A
{{}}                               -----------> Thing B
{{},{{}}}                      -----------> Thing C
{{},{{}},{{},{{}}}}    -----------> Thing D

... and so on.

We choose to give alternative names to these 'things'.

Thing A will be '0'.

Thing B will be '1'.

Thing C will be '2'.

Thing D will be '3'.

... and so on.

As previous posters have already stated, it doesn't really matter what we call each thing.

We could call them apples and bananas; it would just be inconvenient

The 'Essence of Numbers', as you put it, is in the relationship between the sets. The set represented by Thing A is always 'smaller' than Thing B because we can construct Thing B from Thing A, but not the other way around.

[artimies7]

None of these numbers mean anything out of context. Math is just a concept with which to interact with the world around us.

See, 'stupid human minds' created math, so stupid human minds must take it on the same mental plane as it was created on, leaving little room for a reversal on the scale that you are proposing.

If everyone who had ever interacted with mathematics suddenly changed their minds all at once, then yes, you would be correct. As it is now, no.

[agentflare]

Also, principle of explosion.
If 1=0, you can prove anything you want, of course assuming it`s the concept you`re talking about, rather than the arbitrary names.

[bogtro]

Induction would work just fine. We would replace the inductive step with "Assume this is true for n. We show it is true for n-1."

1>0 because we define "1" to be "greater" than "0". A parallel universe with different definitions simply has different definitions.

[northcity4]

I think the thread poster was asking what would happen if our universe got inverted? What if 1 kept it's value as we see it today and it is now <0.

[Dm]

Words are nothing but sounds. We defined them to mean what we want.

I think the thread poster was asking what would happen if our universe got inverted? What if 1 kept it's value as we see it today and it is now <0.

As with most things in life, we would learn to "deal with it" and adapt to the "new" numbers.