Written, in part, to amend an earlier erroneous proof. I more precisely define a theorem;
'There exists a greater structure of which we have a perception of but cannot fully conceive.'
If we could fully conceive such a structure, then we would either understand it as singularity or identify ourselves with it. The structure would be contained within the mind. To prove that we can perceive of a superstructure is to prove that we can come to some understanding of the singularity via its forms, and more strongly for the discussion to be meaningful we would also need to prove that one can conceive of this perception.
The beginning of our proof may prove fruitful to the empirical skeptics who find difficulty in ascertaining the true notion of the infinite, whose necessary existence may not be evident when examining the universe from an epistemology of materialistic construction. In a flavour similar to the ordinal hierarchy we can start from some 'empty set' which here need not embody the abstract notion of set theoretic structure requisite to the continuation of said structure throughout the hierarchy. The reason for this shall become more clear later.
Start with some form X1 and continually generalise it in a sequence X1, X2 , ..., Xn, ...
We shall produce a structure greater than this sequence by constructing a singularity of the sequence itself. We must be careful to divide this proof into various cases. It may be that each generalisation is meaningful and a transfinite sequence is produced, thus there must be some initial singularity X which produces each Xn as its substructure. However the case in which we are truly interested is that when the Xn's are not meaningful and are ontologically null extensions of X1. In this case we cannot come to conceive of the singularity of the higher sequence but we can guarantee its existence by conceiving of X1. This is true by necessity since if the conception of X1 produces X1 again then it must be a singularity, and since by premise we supposed it to be known to the mind it has been conceived. The fundamental equivalence does the rest.
The theorem can know be generalised even further;
'For any known structure one can always guarantee the existence of a structure greater.'
The above argument generalises to any subject matter upon which meaning is being ordained. From the unmeaningful we can produce ontological extensions assuming nothing of their validity, (they very may well be null) but the case holds that the extensions themselves are contributing to a perception of meaning. And the conception of this perception originates from our examination of the initial subject matter. It is important to bear in mind that meaning is originating from the singularity and not the form, forms may well be meaningless and I do not doubt or contradict this at all.
To prove the primary theorem we set X to be the singularity of the self. This is the only case in which ontologically null extensions come into effect since no thought greater than the self can be conceived of. But the existence of higher structure is always necessitated by the theorem and so structure greater than oneself is realised.
Saturday, 15 November 2014
Saturday, 26 July 2014
The Fundamental Equivalence
Examine the existence of a given singularity, and a perception of this singularity.
Fundamental Equivalence - Conceiving of the perception is equivalent to stating the higher singularity necessarily exists.
The proof for the downward part of this theorem is fairly self-evident, to understand the higher singularity is to understand the totality of all its form as an infinitely generalized state. The understanding takes place as a conception from which all such perceptions stem forth. It is as such necessary that one is able to conceive of the perception otherwise the sum totality would not be realised.
The most useful aspect of this realisation is the construction of higher singularity from the conception of its perception. The perception itself must satisfy each higher existence property analogous to the upward Lowenheim-Skolem theorem; though each extension may be ontologically null. The 'higher existence property' is specifically perceiving of the aspect satisfying an order of existence of arbitrarily large size, while this may seem counter-intuitive and indeed paradoxical it is a necessary entailment from the iteration of the aspect to the degree of the cardinality. This is necessary since the singularity by definition is a state of infinite generalisation, and for a perception of this to fully satisfy the sum totality it must satisfy each indexed order of magnitude. This extension is a reflection of the singularity, and not an innate projection from the perception - so it is indeed that the extension from the point of the perception is 'ontologically null', i.e. no new meaning is derived from the construction. From what principle is the meaning ordained? The answer is simple, the singularity.
This aforementioned discussion is now sufficient for the proof of the second direction of the fundamental equivalence. To say that one has knowledge of the higher singularity is to say that one understands the totality of all its form as an infinitely generalised state. The conception of a perception stipulates an infinite generalisation of that aspect, we stated before that the extension is a projection of the singularity rather than the form since arbitrary extension does not generate meaning. To elaborate on this idea consider earlier discussions where it was explained that the 'shadow' as perception exists as that perception only because we relate it to the higher conception. The shadow without knowledge of the conception is meaningless. But the perception itself must be meaningful as being generalised, the generalisation reflects the identity of the higher singularity which henceforth guarantees its existence.
In fact an alternative proof of this account can be given. The conception of a perception lies outside of the immediate field of understanding. To attain this level of generalisation and totality it is clear that one must transcend the immediate circumstance to a higher plane of understanding. The externalisation of this aspect is already apparent, in generalising a perception to conception it is analogous to and a reflective understanding of the generalisation of form to singularity. We can in fact, conceive of the generalisation by abstracting each form. The part cannot exist independent of the whole, and a particular generalisation cannot exist without the embodying conception of totality itself realised as singularity. The subtlety in this argument is not apparent, but we are essentially generalising the generalisation.
One may ask as to why we do not initially invoke the premise that the part cannot exist without the whole to derive the fundamental equivalence. Here one would be making an error to equate the 'whole' with the higher singularity, when the 'whole' could embody the primary conception. That is to say we may have immediate knowledge of the form but nothing higher than that. To prove that the higher singularity exists we could conceive of the form, but it may only produce ontologically null extensions. However the fundamental equivalence should still allow us to guarantee the existence of the higher singularity even though one cannot conceive of it. In practice there is only one situation in which this principle comes into affect, and that is in proving the existence of greater structure to the self.
Fundamental Equivalence - Conceiving of the perception is equivalent to stating the higher singularity necessarily exists.
The proof for the downward part of this theorem is fairly self-evident, to understand the higher singularity is to understand the totality of all its form as an infinitely generalized state. The understanding takes place as a conception from which all such perceptions stem forth. It is as such necessary that one is able to conceive of the perception otherwise the sum totality would not be realised.
The most useful aspect of this realisation is the construction of higher singularity from the conception of its perception. The perception itself must satisfy each higher existence property analogous to the upward Lowenheim-Skolem theorem; though each extension may be ontologically null. The 'higher existence property' is specifically perceiving of the aspect satisfying an order of existence of arbitrarily large size, while this may seem counter-intuitive and indeed paradoxical it is a necessary entailment from the iteration of the aspect to the degree of the cardinality. This is necessary since the singularity by definition is a state of infinite generalisation, and for a perception of this to fully satisfy the sum totality it must satisfy each indexed order of magnitude. This extension is a reflection of the singularity, and not an innate projection from the perception - so it is indeed that the extension from the point of the perception is 'ontologically null', i.e. no new meaning is derived from the construction. From what principle is the meaning ordained? The answer is simple, the singularity.
This aforementioned discussion is now sufficient for the proof of the second direction of the fundamental equivalence. To say that one has knowledge of the higher singularity is to say that one understands the totality of all its form as an infinitely generalised state. The conception of a perception stipulates an infinite generalisation of that aspect, we stated before that the extension is a projection of the singularity rather than the form since arbitrary extension does not generate meaning. To elaborate on this idea consider earlier discussions where it was explained that the 'shadow' as perception exists as that perception only because we relate it to the higher conception. The shadow without knowledge of the conception is meaningless. But the perception itself must be meaningful as being generalised, the generalisation reflects the identity of the higher singularity which henceforth guarantees its existence.
In fact an alternative proof of this account can be given. The conception of a perception lies outside of the immediate field of understanding. To attain this level of generalisation and totality it is clear that one must transcend the immediate circumstance to a higher plane of understanding. The externalisation of this aspect is already apparent, in generalising a perception to conception it is analogous to and a reflective understanding of the generalisation of form to singularity. We can in fact, conceive of the generalisation by abstracting each form. The part cannot exist independent of the whole, and a particular generalisation cannot exist without the embodying conception of totality itself realised as singularity. The subtlety in this argument is not apparent, but we are essentially generalising the generalisation.
One may ask as to why we do not initially invoke the premise that the part cannot exist without the whole to derive the fundamental equivalence. Here one would be making an error to equate the 'whole' with the higher singularity, when the 'whole' could embody the primary conception. That is to say we may have immediate knowledge of the form but nothing higher than that. To prove that the higher singularity exists we could conceive of the form, but it may only produce ontologically null extensions. However the fundamental equivalence should still allow us to guarantee the existence of the higher singularity even though one cannot conceive of it. In practice there is only one situation in which this principle comes into affect, and that is in proving the existence of greater structure to the self.
Thursday, 26 June 2014
The Theory of Perception
A perception, is a particular understanding of some whole, corresponding to a form of the singularity.
The conception of substance is the absolute totality of all forms, the singularity itself.
Skolem's paradox is a result of the downward Lowenheim-Skolem theorem, stating that a countable first order theory has a countable model. The theory of set theory has a countable model, though it necessitates the existence of uncountable sets.
Let us take set theory as the domain of discourse and examine the conception of uncountability of which a countable set is a form. We are not saying that a countable set is a form of an uncountable set, but of the idea of uncountability. It serves as a realisation of an uncountable set, a perception. To understand the issue of uncountability one must harness understanding of all forms, accumulating the understanding of a totality of perception. There may be uncountable perceptions in which case this would fully realise the issue at hand. There may be particular aspects of perception uncountable themselves. The understanding of the structure of the singularity-form principle is long and tenuous; one cannot ever hope to fully realise each particular branch but only the sum union of all such branches. In fact isolating a branch is dangerous, the model or perception may be countable, in this case, which serves as no contradiction because the issue of countability does in fact constitute the identity of uncountability which must be treated as a structural extension rather than a propositional negation.
Human perception may 'fail', but it allows us to percieve of that which is a superstructure to the immediate belief. It is in fact possible to conceive of a perception, which would require knowing all such sub-structural perceptions, whether they may be different realisations in different logical system, or to some other context. Such perception can never 'fail'.
I elaborate further on previous discourse by analogy, suppose there is one who sees the shadow of an object and identifies it as a particular essence of the initial object as singularity, though the shadows does not constitute the structure of the object at all. In fact the shadow has nothing to do with the object but that we had a preconception relating the two. The shadow still identifies with the projection of some being of the object in question. Take the shadow to be a countable set and the object to be the understanding of uncountability. They are only related in a priori knowledge to this degree that countable sets can be used to construct uncountable sets. Take away the a priori knowledge and the shadow crumbles, it cannot possibly be recognised as any realisation of uncountability (which incidentally would be unbeknownst to ourselves). After all, can we blade the shadow for providing a false outlook on the subject matter, it was our perception to fault - the shadows professed no more than what it is.
The conception of substance is the absolute totality of all forms, the singularity itself.
Skolem's paradox is a result of the downward Lowenheim-Skolem theorem, stating that a countable first order theory has a countable model. The theory of set theory has a countable model, though it necessitates the existence of uncountable sets.
Let us take set theory as the domain of discourse and examine the conception of uncountability of which a countable set is a form. We are not saying that a countable set is a form of an uncountable set, but of the idea of uncountability. It serves as a realisation of an uncountable set, a perception. To understand the issue of uncountability one must harness understanding of all forms, accumulating the understanding of a totality of perception. There may be uncountable perceptions in which case this would fully realise the issue at hand. There may be particular aspects of perception uncountable themselves. The understanding of the structure of the singularity-form principle is long and tenuous; one cannot ever hope to fully realise each particular branch but only the sum union of all such branches. In fact isolating a branch is dangerous, the model or perception may be countable, in this case, which serves as no contradiction because the issue of countability does in fact constitute the identity of uncountability which must be treated as a structural extension rather than a propositional negation.
Human perception may 'fail', but it allows us to percieve of that which is a superstructure to the immediate belief. It is in fact possible to conceive of a perception, which would require knowing all such sub-structural perceptions, whether they may be different realisations in different logical system, or to some other context. Such perception can never 'fail'.
I elaborate further on previous discourse by analogy, suppose there is one who sees the shadow of an object and identifies it as a particular essence of the initial object as singularity, though the shadows does not constitute the structure of the object at all. In fact the shadow has nothing to do with the object but that we had a preconception relating the two. The shadow still identifies with the projection of some being of the object in question. Take the shadow to be a countable set and the object to be the understanding of uncountability. They are only related in a priori knowledge to this degree that countable sets can be used to construct uncountable sets. Take away the a priori knowledge and the shadow crumbles, it cannot possibly be recognised as any realisation of uncountability (which incidentally would be unbeknownst to ourselves). After all, can we blade the shadow for providing a false outlook on the subject matter, it was our perception to fault - the shadows professed no more than what it is.
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