The center of mass of your body is a point. The center of mass of your computer is a point as well. There is a distance between these two points. The question is whether this distance is discrete or continuous.A point is an abstract mathematical entity which doesn't correspond with any phenomenon in the world of our everyday existence. — T Clark
Well, that is the subject of discussion.The same is true of a continuum. — T Clark
The center of mass of your body is a point. The center of mass of your computer is a point as well. There is a distance between these two points. The question is whether this distance is discrete or continuous. — MoK
Are you claiming that something which is an abstraction cannot exist?A point does not exist in the everyday world. It is an abstraction, and idealization - imaginary. It has no size. It 's zero dimensional. It does not take up space. A center of gravity is a point and, as such, is also an abstraction, imaginary. — T Clark
Are you claiming that something which is an abstraction cannot exist? — MoK
C2 doesn't follow at all. In the real numbers, there being a gap between 4 and 13 does not imply that the real numbers (or even the rationals) is not a continuum. You need to demonstrate that there is nothing between some pairs of points that are not the same point. Then you've falsified the continuum premise.C1) Therefore there is a gap between all pairs of distinct points of the continuum (from P1 and P2)
C2) Therefore, the continuum does not exist (from A and C1) — MoK
I disagree. Yes, a point can be an abstraction, but can also correspond to a location in space say.A point is an abstract mathematical entity which doesn't correspond with any phenomenon in the world of our everyday existence — T Clark
Only in classical physics, and our universe isn't classical. But I accept your refutation of the rebuttal to the OP. Do you accept my rebuttal?The center of mass of your body is a point. — MoK
A point is an abstract mathematical entity which doesn't correspond with any phenomenon in the world of our everyday existence
— T Clark
I disagree. — noAxioms
C1 states that there is a gap between all pairs of distinct points of the continuum.C2 doesn't follow at all. In the real numbers, there being a gap between 4 and 13 does not imply that the real numbers (or even the rationals) is not a continuum. — noAxioms
Are you challenging (P1)? If yes, I already illustrated that given the definition of the real number one can construct the smallest number or the smallest interval so-called infinitesimal.You need to demonstrate that there is nothing between some pairs of points that are not the same point. Then you've falsified the continuum premise. — noAxioms
You can define it in quantum physics as well. Of course, you cannot measure it.Only in classical physics, and our universe isn't classical. But I accept your refutation of the rebuttal to the OP. Do you accept my rebuttal? — noAxioms
I don't disagree with thatEven if we disagree, the OP still doesn't make sense — T Clark
What do you mean by 'a gap'? If you mean that the two distinct points are not the same point, then yes, by definition. There's a gap between 4 and 13.C1 states that there is a gap between all pairs of distinct points of the continuum. — MoK
Without a definition of a gap, P1 is ambiguous. It states that either G or ~G, which is tautologically true, making P1 empty. The word 'distinct' is not part of P1.Are you challenging (P1)?
Show it then. What about the number that is halfway between this smallest positive number and zero? You've shown that it doesn't exist?It however can be shown that there is the smallest interval on the real number so-called infinitesimal. — MoK
Define it then, without making classical assumptions (like a particle having a location, or some counterfactual property.You can define [the center of mass of a body] it in quantum physics as well. Of course, you cannot measure it.
By a gap, I mean an interval.What do you mean by 'a gap'? If you mean that the two distinct points are not the same point, then yes, by definition. There's a gap between 4 and 13. — noAxioms
It shouldn't be.Without a definition of a gap, P1 is ambiguous. It states that either G or ~G, which is tautologically true, making P1 empty. The word 'distinct' is not part of P1. — noAxioms
Well, I construct the infinitesimal in this way: 0.0...01. By "..." I mean Aleph_0 zero. The next number is then 0.0...02 therefore there is a gap, 0.0...01 between these two numbers. One could say how about 0.0...011? It can be shown that 0.0...011 is 0.0...02 by simple math. 0.0...011=0.0...01+0.0....01. By "...." (where dots appears four times) I mean Aleph_0+1. But Aleph_0+1=Aleph_0 therefore 0.0...01+0.0....01=0.0...01+0.0...01=0.0...02.Show it then. What about the number that is halfway between this smallest positive number and zero? You've shown that it doesn't exist? — noAxioms
This is off-topic but I give it a try. Consider a hydrogen atom for example. R is the center of mass position operator of the atom that is related to the position operator of the nucleus (r_n) and the position operator of the electron (r_e). The relation is R=(m_n*r_n+m_e*r_e)/(m_n+m_e) where m_n and m_e are the mass of the nucleus and electron respectively. The center of mass therefore can be calculated as <Psi(R,t)|R|Psi(R,t)>.Define it then, without making classical assumptions (like a particle having a location, or some counterfactual property. — noAxioms
By a gap, I mean an interval. — MoK
There is however either a gap between all pairs of points of the continuum or there is no gap — MoK
We are dealing with the same point of the continuum if there is no gap between a pair of points — MoK
Therefore there is a gap between all pairs of distinct points of the continuum — MoK
Therefore, the continuum does not exist — MoK
people claim that continuum is the real number. — MoK
The real number, however, is constructed from two parts, an integer part and a decimal part. — MoK
Infinitesimal can be constructed as follows: 0.0...01 by "..." I mean Aleph_0 0 — MoK
I define G2 as an interval between two immediate points with no point between them (what I call an abrupt change in OP). I am interested in understanding whether there are gaps of type G2 in the set of real number given the definition of the set of real number which can be found here. — MoK
infinitesimal_1 = 0.0...01 — MoK
That is not correct: Consider two numbers on the real number such as and . Let's define the mean as . We can determine the next mean as either or in which in the first case we approach to from the right and in the second case we approach to from the left. Let's work with the first approach: . The next mean can then be determined by . We can write . The distance between and is . Therefore, we have . So, . Therefore, your statement does not follow.Between any two distinct real numbers, there is always another one strictly between them. — fishfry
Therefore, we have — MoK
Well, I showed that the distance between consecutive means is zero if the number of divisions is .It appears all you have shown is the distance between consecutive means tends to zero. — jgill
Well, that is the division of two cardinal numbers. I googled about the division of cardinal numbers and I found two references here.The last sentence is a little weird. — jgill
Ok, I see what you mean.The previous sentence says it all if one takes a limit. — jgill
I am still not convinced that infinitesimal does not exist though. Can you prove it? — MoK
I googled and I found two references about the division of cardinal numbers. You can find the references here. — MoK
the distance between consecutive means tends to zero — MoK
Well, I showed that the distance between consecutive means is zero if the number of divisions is aleph_1. — MoK
But the definition isn't constructive and is extensionally unintelligible for some of the reasons you pointed out in the OP. — sime
Dedekind didn't believe in the reality of cuts of the continuum at irrational numbers — sime
so that one never arrives at the antimonies you raised above. — sime
Hmmm. Care to explain? (I recall having difficulty with filters, ultra filters, etc. in grad school a half century ago. I only encountered them in passing - not in my specialty area) — jgill
Ok, thanks for the elaboration. I got that.For a mathematics for the sciences, ordinarily we use a complete ordered field. That requires having a non-empty set, a 2-place relation (<) on the set and two 2-place operations (+ *) on the set such that for all x, y and z:
ORDERED FIELD
x+(y+z) = (x+y)+z (associativity of addition)
x+y = y+x (commutativity of addition)
EyAx x+y = x (additive identity element)
Theorem: E!yAx x+y = x
Definition: 0 = the unique y such Ax x+y = x
Ey x+y = 0 (additive inverse)
EyAx x*y = x (multiplicative identity element)
Theorem: E!yAx x*y = x
Definition: 1 = the unique y such that Ax x*y = x
0 not= 1
x*y = y*x (commutativity of multiplication)
x*(y*z) = (x*y)*z (associativity of multiplication)
x*(y+z) = (x*y)+(x*z) (distributivity)
x not= 0 -> Ey x*y = 1 (multiplicative inverse)
(x < y & y < z) -> x < z (transitivity)
exactly one: x < y, y < x, x = y (trichotomy)
x < y -> x+z < y+z (monotonicity of addition)
(0 < z & x < y) -> x*z < y*z (monotonicity of multiplication)
COMPLETE ORDERED FIELD
In set theory, we prove that there is a carrier set (called 'R') for such a system and such that, for any upper bounded non-empty subset of S of R, S has a least upper bound. With that and the rest of the set theory axioms we can do the mathematics of derivatives and integrals for the sciences. — TonesInDeepFreeze
I looked at all your posts and didn't find the proof that no non-zero real number is an infinitesimal. Could you please provide the proof?An alternative is to have a system with infinitesimals. But still, ordinarily, we need to define <, + and * and to prove whatever theorems are needed for the machinery of mathematics.
To just wave a hand and say "Voila, this is my infinitesimal" does not provide the needed definitions of < + and * with infinitesimals nor the needed proofs.
So how do we go about proving the existence of a system with infinitesimals? For your purposes, it would help to first define 'is an infinitesimal'. I provided a definition previously, but I notice that many authors include 0 as an infinitesimal. So perhaps use this definition:
x is an infinitesimal if and only if, for every positive real number y, |x| < y.
It has been proven for you that for every real number x there is a positive real number y such that y < |x|.
So no non-zero real number is an infinitesimal.
One more time: No non-zero real number is an infinitesimal. The proof that no non-zero real number is an infinitesimal is immediate from the fact that for every real number x there is a positive real number y such that y < |x|. We don't need to keep going over this over and over. — TonesInDeepFreeze
The sequence of half distances converges to 0. So what? That doesn't prove that it's not the case that between any two different real numbers there is another different real number. — TonesInDeepFreeze
This was a reply to the above comment from @fishfry who claimed between any two distinct real numbers, there is always another one strictly between. The distance between two points is zero if the number of divisions is strictly infinite so there cannot be a point between two points in this case.Between any two distinct real numbers, there is always another one strictly between them. — fishfry
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