mersenneforum.org The quantity of integers distributed along the number line is an odd quantity
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 2022-02-01, 19:23 #1 Charles Kusniec     Aug 2020 Guarujá - Brasil 59 Posts The quantity of integers distributed along the number line is an odd quantity In the XY-plane we have the X-axis and the Y-axis, each with a number line containing all integers. So, if we make the bijection between the positive numbers on the X and Y axis, then we can say that the number of positive elements on both axes is an even number. The same is true for negative integers. So, if we now consider that zero has no corresponding pair, then we can state that the quantity of integers distributed along the X and Y axes in 2D of the XY plane is an odd quantity. Similarly, this reasoning can be done for the 1D number line and also for 3D space. All of them have an odd number of integers distributed along their axes. Is this reasoning correct? Does anyone see anything that has not been considered? Thank you,
2022-02-01, 20:04   #2
Dr Sardonicus

Feb 2017
Nowhere

3×11×181 Posts

Quote:
 Originally Posted by Charles Kusniec In the XY-plane we have the X-axis and the Y-axis, each with a number line containing all integers. So, if we make the bijection between the positive numbers on the X and Y axis, then we can say that the number of positive elements on both axes is an even number.
Doubly wrong. The number of positive integers is not an integer, so the terms "even number" and "odd number" do not apply.

Further, "positive numbers" and "positive elements" on a coordinate axis means positive real numbers, not just integers.
Quote:
 Is this reasoning correct?
No. You don't even know the definitions of the terms you're using.

2022-02-01, 20:07   #3
paulunderwood

Sep 2002
Database er0rr

2×3×23×31 Posts

Quote:
 then we can say that the number of positive elements on both axes is an even number.
The integers have cardinality $$\aleph_0$$. As do positive integers. Negative too. Odd numbers too and even numbers. And prime numbers. All have cardinality $$\aleph_0$$. As do the integer points on your grid. The same is true for the axes with integer points.

Last fiddled with by paulunderwood on 2022-02-01 at 20:18

2022-02-01, 20:59   #4
VBCurtis

"Curtis"
Feb 2005
Riverside, CA

5,471 Posts

Quote:
 Originally Posted by Charles Kusniec So, if we make the bijection between the positive numbers on the X and Y axis, then we can say that the number of positive elements on both axes is an even number. The same is true for negative integers.
Why can you say it is even? What does "the bijection" have to do with it? Be specific here- you're reaching meaningless conclusions because you're waving your hands without considering the meaning of the words you're using.

2022-02-01, 22:41   #5
Charles Kusniec

Aug 2020
Guarujá - Brasil

738 Posts

Quote:
 Originally Posted by paulunderwood The integers have cardinality $$\aleph_0$$. As do positive integers. Negative too. Odd numbers too and even numbers. And prime numbers. All have cardinality $$\aleph_0$$. As do the integer points on your grid. The same is true for the axes with integer points.
Excluding the number 0, the cardinality A of positive integers is equal to the cardinality B of negative integers: A=B.
Now, can the cardinality of all positive and negative integers (A+B) without the number zero, be considered as an even cardinality?

 2022-02-01, 22:45 #6 a1call     "Rashid Naimi" Oct 2015 Remote to Here/There 8EF16 Posts Even numbers can be bisected to even numbers. Any way you look at it it does not follow that the number of positive integers is odd (or even for that matter). Last fiddled with by a1call on 2022-02-01 at 22:46
2022-02-01, 22:48   #7
paulunderwood

Sep 2002
Database er0rr

427810 Posts

Quote:
 Originally Posted by Charles Kusniec Excluding the number 0, the cardinality A of positive integers is equal to the cardinality B of negative integers: A=B. Now, can the cardinality of all positive and negative integers (A+B) without the number zero, be considered as an even cardinality?
$$\aleph_0 + 1 = \aleph_0$$, so neither odd nor even.

Also $$\aleph_0 + \aleph_0\ = \aleph_0$$.

See: https://en.wikipedia.org/wiki/Cardinality

2022-02-01, 22:56   #8
a1call

"Rashid Naimi"
Oct 2015
Remote to Here/There

2,287 Posts

Quote:
 Originally Posted by Charles Kusniec Excluding the number 0, the cardinality A of positive integers is equal to the cardinality B of negative integers: A=B. Now, can the cardinality of all positive and negative integers (A+B) without the number zero, be considered as an even cardinality?
I think what you are trying to say is:
There is a one-to-one relationship between all positive integers to all negative integers and 0 is the tie-breaker making an odd total.
While that argument works for finite sets, it does not follow that it will hold true for infinite sets. There is a one-to one relationship between all countable infinite sets. So there is a one-to-one relationship between any infinite subset of integers (say all multiples of 19) and complete set of integers. Normal rationales break down when dealing with infinites.

2022-02-01, 23:42   #9
Dr Sardonicus

Feb 2017
Nowhere

3·11·181 Posts

Quote:
 Originally Posted by Charles Kusniec Excluding the number 0, the cardinality A of positive integers is equal to the cardinality B of negative integers: A=B. Now, can the cardinality of all positive and negative integers (A+B) without the number zero, be considered as an even cardinality?
No. I've already explained. So have others. You're obviously not paying any attention.

2022-02-02, 00:22   #10
VBCurtis

"Curtis"
Feb 2005
Riverside, CA

5,471 Posts

Quote:
 Originally Posted by Charles Kusniec ...even cardinality?
Define this phrase for an infinite set. Since you haven't done that, your question is meaningless.

 2022-02-02, 03:43 #11 retina Undefined     "The unspeakable one" Jun 2006 My evil lair 5·1,319 Posts Isn't there an unspoken assumption that the axes must always cross through (0,0)? I could just as (in)correctly state that all axes cross at (1,1), thus zero has two instances on each of the X and Y axes. And that since one is common to both axes then there only a single one, so there are fewer odd numbers than even numbers. Indeed there are an infinite number of places we can put the crossing point of each axis. Crossing at (0,0) is an arbitrary choice. BTW I had to write two zeros (0,0) when defining the point. So does that mean there is more than one zero anyway? Last fiddled with by retina on 2022-02-02 at 03:47

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