- 868

- 3

## Main Question or Discussion Point

My skill with proofs and number theory is very limited and my use tends toward the stereotyped physicist. Dealing with infinities and identifying exactly what assumptions are in play has become somewhat an issue for me though. My questions are contained in the following visual tool.

Please slap me in line if I mis-use any terms or logic.

Define two line segments [tex]l_1[/tex] and [tex]l_2[/tex], y=mx + b, where the slopes are 0.

[tex]l_1[/tex] where [tex]b_1 = 2[/tex] and [tex]2<y_1<-2[/tex]

[tex]l_2[/tex] where [tex]b_2 = -1[/tex] and [tex]1<y_2<-1[/tex]

Let set [tex]{R_1}[/tex] be a subset containing all points represented by one and only one real number, cardinality 1. Dito for [tex]R_2[/tex].

Theorems:

1) The number of members of the set [tex]R_1[/tex] or [tex]R_2[/tex] is infinite.

2) Origin bijection - For each line that intercepts [tex]l_1, l_2,[/tex] and point (0,0) there exist one and only one member each of [tex]R_1[/tex] and [tex]R_2[/tex] that this same line intercepts both [tex]l_1[/tex] and [tex]l_2[/tex], i.e., not surjective.

3) Origin bijection is a real number logical equivalent of the integer bijection used by Cantor in transfinite sets where

[tex]l_1 \equiv \sum_{n=1}^\infty n[/tex]

and

[tex]l_2 \equiv \sum_{n=1}^\infty 2n[/tex]

using infinitesimals rather than limits.

Question: What prior art exist similar to this notion of origin bijection?

Implications:

1) Any finite one to one mapping via origin bijection indicates that [tex]l_2/l_1 = 1[/tex], yet in the limit must equal 1/2 in this case, and varies generally.

2) Implication 1) indicates the set of infinitesimals not contained in the subset R, which are neither measurable nor countable, must have a total magnitude (summation) equal to the total magnitude of the subset R defined in l.

3) We can maintain that implication 2) is not due to a summation of infinitesimals not contained in the subset R but that indicates that 1 to 1 correspondence is invalid. Vitali sets specifically defines this as a property of reals.

4) We could assume the difference in [tex]l_1[/tex] and [tex]l_2[/tex] are a result of irrational numbers. This implies that the ratio of irrational numbers to real numbers can take any value, finite and otherwise, i.e., the irrational numbers defined on any line segment can sum to any value we choose simply by choice of scale.

5) Scale freedom (coordinate independence) is a fundamental principle and we could assume that the ratio of reals to irrationals has some actual value that changes due to scale. However, if this was the case, in principle the same bijection effects should apply to the irrationals.

6) If we reject this scale choice as a property of irrationals then we must assume a set of points not defined by any definable number must contain this property of scale freedom, i.e., not Lebesgue measurable. Assuming this is a property of irrationals means that the axiom of choice is the only assumption required to say 1.99... = 2. Defining how and where these non-measurable sets act on scale appears as a moving target and allows choice in how to define them.

Axiom of choice & Lebesgue measure:

Assuming AC not all subsets of a space are Lebesgue measurable. Wrt origin bijection the Lebesgue immeasurability (non-measurable sets) implicit in AC appears as the infinitesimals that are not accounted for in the 1 to 1 origin bijection. This is the property I identified above as scale freedom. This implies that sets which are not Lebesgue measurable may use scale freedom to redefine spaces of one magnitude to spaces of another magnitude. A related case can be made that scale freedom requires this property of non-measurable sets to be mathematically consistent. How would a choice of scale remain mathematically consistant without non-measurable sets with which to define this choice? The non-measurable sets in [tex]l_1[/tex] and [tex]l_2[/tex] play the same role whether it is a change of scale or simply a redefinition of scale where we simple choose to define length [tex]l_1 = 1[/tex].

Conjecture - Lebesgue immeasurable sets and scale freedom or choice is equivalent.

Banach-Tarski theorem & Vitali sets:

Both Banach-Tarski theorem and Vitali sets depend critically on AC and the associated construction of non-measurable sets. Vitali sets define these non-measurable sets as reals yet remain impossible to describe explicitly. If the above conjecture holds then these theorems simply redefines scale rather than construct one volume from another. So the choice in ac that requires non-measurable sets is a choice of scale, not a change of scale, by the above conjecture.

Questions:

Ive seen some references to attempts via nonconstructive proof to exploit these properties of non-measurable sets via AC. What prior work exist that attempts to resolve the paradoxes without rejecting AC? What other theorems etc. might be related to this I should be made aware of? What prior work is similar to the concept of origin bijection I used here? What objections and/or logical errors can be pointed out here?

Please slap me in line if I mis-use any terms or logic.

Define two line segments [tex]l_1[/tex] and [tex]l_2[/tex], y=mx + b, where the slopes are 0.

[tex]l_1[/tex] where [tex]b_1 = 2[/tex] and [tex]2<y_1<-2[/tex]

[tex]l_2[/tex] where [tex]b_2 = -1[/tex] and [tex]1<y_2<-1[/tex]

Let set [tex]{R_1}[/tex] be a subset containing all points represented by one and only one real number, cardinality 1. Dito for [tex]R_2[/tex].

Theorems:

1) The number of members of the set [tex]R_1[/tex] or [tex]R_2[/tex] is infinite.

2) Origin bijection - For each line that intercepts [tex]l_1, l_2,[/tex] and point (0,0) there exist one and only one member each of [tex]R_1[/tex] and [tex]R_2[/tex] that this same line intercepts both [tex]l_1[/tex] and [tex]l_2[/tex], i.e., not surjective.

3) Origin bijection is a real number logical equivalent of the integer bijection used by Cantor in transfinite sets where

[tex]l_1 \equiv \sum_{n=1}^\infty n[/tex]

and

[tex]l_2 \equiv \sum_{n=1}^\infty 2n[/tex]

using infinitesimals rather than limits.

Question: What prior art exist similar to this notion of origin bijection?

Implications:

1) Any finite one to one mapping via origin bijection indicates that [tex]l_2/l_1 = 1[/tex], yet in the limit must equal 1/2 in this case, and varies generally.

2) Implication 1) indicates the set of infinitesimals not contained in the subset R, which are neither measurable nor countable, must have a total magnitude (summation) equal to the total magnitude of the subset R defined in l.

3) We can maintain that implication 2) is not due to a summation of infinitesimals not contained in the subset R but that indicates that 1 to 1 correspondence is invalid. Vitali sets specifically defines this as a property of reals.

4) We could assume the difference in [tex]l_1[/tex] and [tex]l_2[/tex] are a result of irrational numbers. This implies that the ratio of irrational numbers to real numbers can take any value, finite and otherwise, i.e., the irrational numbers defined on any line segment can sum to any value we choose simply by choice of scale.

5) Scale freedom (coordinate independence) is a fundamental principle and we could assume that the ratio of reals to irrationals has some actual value that changes due to scale. However, if this was the case, in principle the same bijection effects should apply to the irrationals.

6) If we reject this scale choice as a property of irrationals then we must assume a set of points not defined by any definable number must contain this property of scale freedom, i.e., not Lebesgue measurable. Assuming this is a property of irrationals means that the axiom of choice is the only assumption required to say 1.99... = 2. Defining how and where these non-measurable sets act on scale appears as a moving target and allows choice in how to define them.

Axiom of choice & Lebesgue measure:

Assuming AC not all subsets of a space are Lebesgue measurable. Wrt origin bijection the Lebesgue immeasurability (non-measurable sets) implicit in AC appears as the infinitesimals that are not accounted for in the 1 to 1 origin bijection. This is the property I identified above as scale freedom. This implies that sets which are not Lebesgue measurable may use scale freedom to redefine spaces of one magnitude to spaces of another magnitude. A related case can be made that scale freedom requires this property of non-measurable sets to be mathematically consistent. How would a choice of scale remain mathematically consistant without non-measurable sets with which to define this choice? The non-measurable sets in [tex]l_1[/tex] and [tex]l_2[/tex] play the same role whether it is a change of scale or simply a redefinition of scale where we simple choose to define length [tex]l_1 = 1[/tex].

Conjecture - Lebesgue immeasurable sets and scale freedom or choice is equivalent.

Banach-Tarski theorem & Vitali sets:

Both Banach-Tarski theorem and Vitali sets depend critically on AC and the associated construction of non-measurable sets. Vitali sets define these non-measurable sets as reals yet remain impossible to describe explicitly. If the above conjecture holds then these theorems simply redefines scale rather than construct one volume from another. So the choice in ac that requires non-measurable sets is a choice of scale, not a change of scale, by the above conjecture.

Questions:

Ive seen some references to attempts via nonconstructive proof to exploit these properties of non-measurable sets via AC. What prior work exist that attempts to resolve the paradoxes without rejecting AC? What other theorems etc. might be related to this I should be made aware of? What prior work is similar to the concept of origin bijection I used here? What objections and/or logical errors can be pointed out here?