How Paradox Extends Logic
Russells paradox reveals a contradiction embedded within the first set-theoretical axiom of math.
The unrestricted axiom of comprehension in set theory states that to every condition there corresponds a set of things meeting the condition: (?y) (y={x : Fx}). The axiom needs restriction, since Russell's paradox shows that in this form it will lead to contradiction.
In mathematical logic, Russell's paradox is a set-theoretic paradox published by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains an unrestricted comprehension principle leads to contradictions. Wikipedia
Let R = {x | x ? x}, then R ? R ? R ? R
This OP will argue that the contradiction embedded in the unrestricted axiom of comprehension is not a fatal flaw disfiguring the relationship between math and set theory, and that unrestricted comprehension of set membership is achievable, albeit in stages.
Premise paradox equals inconsistent equality
Premise a paradox is a signpost pointing towards higher-order dimensional expansion
Premise unrestricted comprehension is conditionally limited within a series of well-defined contexts
Premise every complex of inter-connected dimensions generates paradoxes
For my argument by example, I will use our everyday world of three spatial dimensions.
Unspecifiable existence, in its effort to get itself into measurability, expands. At the level
of zero dimensions the singularity axiomatic disequilibrium sparks time, energy, motion and space into expansion beyond the unspecifiably-existent singularity.
Unspecifiable existence, in its effort to get itself into measurability, disappears. This is the first paradox within our universe of three empirical spatial dimensions.
Singularity-as-point. ? Unspecifiable existence. Zero dimensions.
First Paradox: Point-as-line ? The singularity is there and not there. One dimension.
Second Paradox: Line-as-arc converges with 360 degree sphere. ?The line as circle. Two dimensions.
Third Paradox: Parallelogram-as-hemisphere converges with diameter. ? The circle as sphere. Three dimensions.
Fourth Paradox: Cube as time-zero expanding universe converges with hypercubic space. ? The sphere as hypercube (Russells Paradox). Four dimensions.
The dimensionally-expanded universe is an open, bounded infinity that graduates in steps with paradox binders acting as the stitching.
In the context of open, bounded infinity configured in graduated steps with paradox binders, logic is a continuity of motion-as-empirical-narrative that evolves toward the self-reference of a borderline with next upward step of expanded dimensionality.
Paradox tells us weve reached a logical limit wherein a higher-order dimension in collapsed configuration populates the paradox.
Expansion of the higher-order dimension resolves the paradox.
When Frege and Russell confronted the paradox inherent within the unrestricted-comprehension axiom of set theory, they were gazing upon the infra-structure of an expandable universe configured in steps.
They were looking at metaphysics itself.
The metaphysics of a step-hierarchical universe is physical transcendence in steps bounded by paradox.
Russells paradox understood as paradox-to-be-articulated as expansion of a collapsed higher-order dimension becomes a defense of physical transcendence and therefore of transcendental metaphysics.
Transcendental Logic ? Transcendental Metaphysics
The unrestricted axiom of comprehension in set theory states that to every condition there corresponds a set of things meeting the condition: (?y) (y={x : Fx}). The axiom needs restriction, since Russell's paradox shows that in this form it will lead to contradiction.
In mathematical logic, Russell's paradox is a set-theoretic paradox published by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains an unrestricted comprehension principle leads to contradictions. Wikipedia
Let R = {x | x ? x}, then R ? R ? R ? R
This OP will argue that the contradiction embedded in the unrestricted axiom of comprehension is not a fatal flaw disfiguring the relationship between math and set theory, and that unrestricted comprehension of set membership is achievable, albeit in stages.
Premise paradox equals inconsistent equality
Premise a paradox is a signpost pointing towards higher-order dimensional expansion
Premise unrestricted comprehension is conditionally limited within a series of well-defined contexts
Premise every complex of inter-connected dimensions generates paradoxes
For my argument by example, I will use our everyday world of three spatial dimensions.
Unspecifiable existence, in its effort to get itself into measurability, expands. At the level
of zero dimensions the singularity axiomatic disequilibrium sparks time, energy, motion and space into expansion beyond the unspecifiably-existent singularity.
Unspecifiable existence, in its effort to get itself into measurability, disappears. This is the first paradox within our universe of three empirical spatial dimensions.
Singularity-as-point. ? Unspecifiable existence. Zero dimensions.
First Paradox: Point-as-line ? The singularity is there and not there. One dimension.
Second Paradox: Line-as-arc converges with 360 degree sphere. ?The line as circle. Two dimensions.
Third Paradox: Parallelogram-as-hemisphere converges with diameter. ? The circle as sphere. Three dimensions.
Fourth Paradox: Cube as time-zero expanding universe converges with hypercubic space. ? The sphere as hypercube (Russells Paradox). Four dimensions.
The dimensionally-expanded universe is an open, bounded infinity that graduates in steps with paradox binders acting as the stitching.
In the context of open, bounded infinity configured in graduated steps with paradox binders, logic is a continuity of motion-as-empirical-narrative that evolves toward the self-reference of a borderline with next upward step of expanded dimensionality.
Paradox tells us weve reached a logical limit wherein a higher-order dimension in collapsed configuration populates the paradox.
Expansion of the higher-order dimension resolves the paradox.
When Frege and Russell confronted the paradox inherent within the unrestricted-comprehension axiom of set theory, they were gazing upon the infra-structure of an expandable universe configured in steps.
- Let R = {x | x ? x}, then R ? R ? R ? R (The set of all sets not members of themselves)
- With upward dimensional expansion to hypercubic space, the paradox is resolved because hypercubic space, being bounded by cubic space, logically occupies with consistency at 4D what is paradoxical at 3D. In other words, hypercubic space, when viewed through a 3D lens, occupies two places at once.
They were looking at metaphysics itself.
The metaphysics of a step-hierarchical universe is physical transcendence in steps bounded by paradox.
Russells paradox understood as paradox-to-be-articulated as expansion of a collapsed higher-order dimension becomes a defense of physical transcendence and therefore of transcendental metaphysics.
Transcendental Logic ? Transcendental Metaphysics
Comments (34)
Qu'est-ce qui ne va pas, c'est facile à voir?
A set is just a collection of any type of things.
What are dimensions doing in set theory?
A vector space is a set of "objects" whose "dimension" is the cardinality of its basis. But this is linear algebra rather than purer set theory. You made a good point. :up:
Quoting Banno
With the above help from jgill, I acknowledge the authority of your point, Banno.
Frege and Russell wanted to reduce math to {first-order logic + set theory} by declaring that numbers-as-numbers are sets: the number 4, for example is a set.
The first rule of set theory: unrestricted comprehension, Russell showed, leads to paradox.
The set-theoretical rule of restricted comprehension, an adjustment configured by Russell and others, aims to return math and set theory to consistency.
Kaplan, in the video for the first link in my OP, argues that mathematicians cannot effect this return to consistency. He shows that the predication of grammatical logic, like the set-theoretical logic of math, ends in the same paradox. He adds that working around it with restricted boundaries declared by fiat does nothing to change this.
The gist of my OP is my argument for recognizing the first-order logical consistency of unrestricted comprehension by utilizing the ascending sequence of dimensional complexes as steps that collectively establish said consistency.
The crux of the steps argument is the premise that paradox is the binder that connects the steps and preserves consistency across them.
If each step is a domain, then the boundary of a given domain is reached when a boundary definition, such as the set of all sets not members of themselves, contains paradox. The paradox tells us we have reached a dimension of higher-order than the scope of the domain and therefore, to preserve consistency, we must expand the collapsed dimension. This expansion moves us up to the next higher domain (step) of dimensional expansion. Within this higher domain, the paradox of the previous domain is resolved by expansion of the previously collapsed dimension. For example: cubic space being in two places at once is paradox whereas hypercubic space being in two places at once is not.
Preserving the consistency of unrestricted comprehension, as you may have noticed, resembles the technique by which calculus makes approximations of negligible imprecision of irrational dimensions such as the area under a curve.
What's that?
Unrestricted comprehension within the domain of 3D leads to paradox: inconsistency. Unrestricted comprehension across the duet of 3D_4D leads to expansion of hypercubic space with preservation of consistency. No paradox and no need to rejigger the rule.
While we're talking about it, got any idea what a 4D paradox looks like?
What could unrestricted comprehension be? Comprehension is not a term in set theory.
For example, the set of all setsthe universal setwould be {x | x = x}.
Central to my argument is: the set of all sets not members of themselves.
Quoting Banno
I'm lost here, too. :roll:
:up: That's a good point mon ami! Not always though and hence your thread, oui?
That was avoided in ZFC by the separation axiom.
So back to my original question, what are dimensions doing in set theory? What is a dimension here?
You got it! Yes. That's the gist of my argument.
:lol: I'm not sure how exactly though.
Quoting Agent Smith
Quoting ucarr
Quoting Banno
Quoting ucarr
This set, as shown by Russell, leads to a paradoxical conclusion such that the set of all sets not members of themselves is simultaneously a member of itself and not a member of itself.
We can take this paradox and cast it into another, equivalent form: being in two places at the same time which means an object is simultaneously itself and not itself.
Since a hypercube, being 4D, has 3D boundaries, it occupies four distant 3D locations, i.e., the same object in four places simultaneously. This type of spatial expansion, i.e., spatial dimension, deals a fatal blow to logical consistency at the level of 3D spatial expansion. At the level of 4D spatial expansion, logical consistency, i.e., one object being in two places at once is natural not fatal.
From these ruminations we see clearly the direct linkage binding logic and spatial dimensions. Conceptually speaking, logic, which is continuity, concerns itself figuratively with dimensional expansion in the form of an expansion of logical inferences.
Physically speaking, the logic that grounds the math that measures spatially extended objects, when confronted with simultaneous occupation of two different locations at the level of 3D, descends into paradox. This is, however, a simple case of dimensional expansion (symbolic and literal) butting up against a boundary. My theory argues that said boundary is not impassable. One need simply realize paradox is a signpost signaling a boundary for set theoretical logic within a specific matrix of dimensional expansion. My concomitant theory that our physical universe is configured in dimensional matrices that progress in steps resolves the impasse with recourse to ascension to a higher-dimensional matrix.
If you haven't watched unrestricted axiom of comprehension please humor me and do so. The brilliant Jeffery Kaplan presents a cogent argument declaring that the problem of unrestricted comprehension of sets is ongoing; ZFC has not resolved the problem.
Quoting Agent Smith
If you haven't watched unrestricted axiom of comprehension please humor me and do so. The brilliant Jeffery Kaplan presents a cogent argument declaring that the problem of unrestricted comprehension of sets is ongoing; ZFC has not resolved the problem.
Are you perhaps talking about, say, an interaction between two hypercubes?
I'm afraid I don't understand where the paradox is in 4D hypercubes. Let's simplify for a moment to better visualize the problem. A 2D square has 1D boundaries (lines) in 4 different locations, meeting at the edges. This is the same relationship that a 3D cube has with its 2D sides, and that a 4D hypercube has with its 3D sides.
What is the contradiction in a square having lines in different locations, meeting at the edges? What is the contradiction in a 4D hypercube having 3D sides in different locations, meeting at the edges? And what has that to do with Russel's paradox?
No, definitely not, that kinda stuff is above me pay grade mate, but look at the underlined term in your sentence.
A very stretched metaphor, at best; not an equivalence.
But sure, Russell's paradox lead to further developments in logic, not to its demise.
The parallelism of metaphor and the identity of math are distinct.
You have an identity. I'm guessing you think you cannot be in two locations simultaneously. It goes beyond a real limitation of our 3D reality. If we imagine an instance when you are in two different locations simultaneously within our 3D reality, that means the unique you -- not your and your twin -- is in Location A and the unique you is in Location B. This simultaneity of unique you in two different locations at once compels us to say: you are in Location A and you are not in Location A; you are in Location B and you are not in Location B. If you and not-you are simultaneous, then you are yourself and not yourself; in this example bi-directionally. This is not parallelism. This is paradoxical identity.
Quoting Banno
You're refuting a claim never made.
There's no paradox in 4D hypercubes, 3D cubes, 2D parallelograms, 1D lines, 0D points.
Paradox appears when, for example, a 3D configuration tries to contain a 4D configuration. Frege did this conceptually when he conceived of the set of all sets not members of themselves.
One of the cruxes of my claim is that paradox appears as a symptom of a border crossing by a higher dimensional configuration into the realm of a lower dimensional configuration.
That such boundaries exist between levels of dimensional configurations is evidence that our universe is metaphysically configured in ascending steps of upwardly dimensional configurations.
Note - Meta, as in metaphysical, doesn't mean immaterial_spiritual. It just means higher-order. For example, a meta-narrative, being higher-order than narrative, contains narrative as a subset plus more. So metaphysics herein means a bigger set that contains physics as a subset.
Thank-you. I need your scrutiny. May it continue.
:ok:
Quoting Agent Smith
So far I've gotta wild speculation about what you're suggesting here. Could it be you're suggesting cardinality in 4-space is categorically different from what it is in 3-space? Are we looking at a difference such that hypercubes don't proliferate in the same way cubes proliferate?