Why don't you just address the question rather than avoiding it. You need to provide some proof real numbers have immediate successors and immediate predecessors.
jsfisher, you say:
The use of the word
all doesn't guarantee they exist. As for the "no gaps" characteristic, that's what guarantees they don't exist.
I say:
The use of
all + the fact that distinct
R members have "no gaps" (and both of them are used by Standard Math) guarantees (in the framework of Standard Math) that Y has an immediate predecessor, that cannot be found by using any particuler and finite case of pair of
R members, as used in your "proof".
All you have to do is to show that your "proof" is not about any particular pair of
R members, and only then your "proof" may become a proof.
It is clearly shown that your "proof" cannot avoid particular pair of
R members as an inseparable part of it.
As a result, you have no proof.
If you don't agree with me, then you have to explicitly show why Y has no immediate predecessor by using a particular pair of
R members as an inseparable part of your "proof".
EDIT:
Here is your "proof" once again:
jsfisher said:
Assume the set {X : X<Y} does have a largest element, Z.
For Z to be an element of the set, Z < Y.
Let h be any element of the interval (Z,Y).
By the construction of h, Z < h < Y.
Since h < Y, h is an element of the set {X : X<Y}.
Since Z < h, the assumption Z was the largest element of the set has been contradicted.
Therefore, the set {X : X<Y} does not have a largest element.
If one assumes that the set {X : X<Y} does have a largest element, Z, then one assumes the case [X,Z] or the case [Z,Y), that clearly can't be used to conclude anything about the non existence of the immediate predecessor of Y.
Furthermore, you write:
For Z to be an element of the set, Z < Y.
Let h be any element of the interval (Z,Y).
Jsfisher, if Z is the largest element of [X,Z] such that Z<Y, then there is no room for h between Z and Y, exactly because [X,Y) is an interval of
all R members, and
all R members have no room ("gap") for your h.
In that case, you actually have in your hand the exact value of the immediate predecessor of Y, but when you try to show it, you find yourself closed under the finite case of Z and Y of [Z,Y), which prevents from you to show the exact value of the immediate predecessor of Y of the non-finite collection of
all R members of [X,Y) interval.
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This case is equivalent (by analogy) to the wave\particle measurement problem of QM.
As long as you do not explicitly check the location of the wavicle, its existence is based on a superposition of
all infinitely many locations of a given interval.
At the moment that you check it, only one finite case is found, but by using this particular finite case, you can't conclude anything about the superposition of
all infinitely many locations of a given interval.
