User talk:KBlott

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 * --Your friendly ProofWiki WelcomeBot 22:15, 7 January 2012 (EST)

Subpages
Hey, I've moved some of the pages you've made to be subpages of your user page. --Joe (talk) 16:21, 8 January 2012 (EST)

Your strategy
I've been watching some of your edits from a distance for a while, and I'm interested: what is your strategy? Are you planning on either (a) incorporating it into our general house structure at any stage, and (b) linking it all up with the existing work? Or is all this being designed as a completely independent body of knowledge? --prime mover 01:30, 17 January 2012 (EST)
 * I have no particular strategy really. I just go where the evidence takes me and all work is essentially derivative.
 * Notice that this page is incompatible with standard set theory.  I would not dream of attempting to integrate it into work that is based on standard set theory as it would quickly lead to problems.  There is evidence, however, that the Cardinality of sets can be relative.   Consider the set of all locations of a particle at a given time.  It is generally assumed that such sets are necessarily singleton.  (A particle is assumed to have only one location at a given time.)  However, the double slit experiment refutes this assumption.  This suggests that inclusion in a given set ($\in$) may be relative to the observer and, of course, the time the observation is made ($\in_t$).
 * Class algebra is mentioned on this web site but seems not to be treated with a great deal of reverence. Although mappings obviously operate on classes, their definition on this web site is restricted to sets.
 * The same is true of the definition of lattices on this web site. Notice that the definition of a lattice given by these authors is not equivalent.  While all lattices of sets are lattices of classes, the converse is not true.  Consider, for example the class $U$ of all sets.  This is a class lattice (with $\vee = \cup$ and $\wedge = \cap$) but it is not a set lattice because $U$ is not a set.  --KBlott 16:18, 17 January 2012 (EST)
 * I can see separate interest for what you are developing on referenced page for $\in$. It appears that this yields a rigorous (possibly extendible) definition of $\in$ when considering non-standard set theory and logic (where eg. a lattice of truth values is used; though I don't know if something like that is viable). --Lord_Farin 16:45, 17 January 2012 (EST)
 * Several authors have attempted to reformulate propositional calculus in terms of lattice theory. Two such attempts are cited by Sudbery on page 209.  Though as Sudbury himself pointed out “it would involve reconstructing the whole of mathematics” (p 223).  Who has that much time?  I sure don’t.


 * I do believe that such a reconstruction is necessary. Einstein believed that the laws of physics are invariant under all possible co-ordinate transformations.  However, he restricted his attention to one-to-one co-ordinate transformations and failed to unify his theory of gravity with quantum mechanics.


 * Consider a co-ordinate transformation of the world line of an electron from its own frame of reference to an observer's frame of reference. In the electron's frame of reference, the electron's location is fixed.  So, the mechanics of its motion is very simple.  Now consider the motion of the electron in an observer’s frame of reference.  Let $t$ be the time on the observer’s clock.  Let $S$ be the set of all points in space from the observers point of view.  Let $L \in$ $\mathcal P(S)$ be the set of all points in $S$ at which the electron happens to be located.  Since, the co-ordinate transformation from the electron’s frame of reference to the observer's frame of reference may not be one-to-one, the electron’s world line may map onto a particular point in $S$ more than once.  Worse still, the electron may be moving either forward in time or backward in time, with respect to the observer, at any given instance.  So, we need two sets of natural numbers: $n_+$ to describe the number of times it maps to a point $\underline x  \in S$ while traveling forward in time and $n_-$ to describe the number of times it maps to a point $\underline x  \in S$ while traveling backward in time.  So, rather than having two truth values (true or false), $\underline x \in_t L$ maps to  infinitely many truth values $(n_+, n_-)$.  So, $\in_t: S \times \mathcal P(S) \to \mathbb N \times \mathbb N$.  I was thinking about this when I wrote this page. --KBlott 20:03, 17 January 2012 (EST)
 * What I would like you to do would be to consider posting up some of your axioms and definitions. You may have noticed that an attempt is made on this site to link everything back to axioms (in particular, ZFC). If your work is based on a completely different set of axioms, they need to be stated otherwise nothing you post can be confirmed. --prime mover 01:36, 18 January 2012 (EST)
 * I would be honoured attempt to prove that $\mathbb N \times \mathbb N$ is a lattice. This fact follows from the discussion above.  Since $\mathbb N$ is a set, it should be at least consistent with ZFC.  Though for proper classes I think it will be necessary to modify ZFC.
 * In order to be a lattice, a set needs an ordering of some kind. What ordering do you have in mind? If, as I suspect, you are considering what you call "protointegers", all this has already been done (using different terminology) during the operation to create the integers as an inverse completion of the natural numbers. In that case there is no need to show it's a lattice as it is already established that the conventional ordering is total. By an elementary (nay, trivial) proof, a total ordering is a lattice.
 * Unless the object you are creating is nothing to do with protointegers but is something else altogether, in which case you need to specify exactly what you're talking about. As it is, I can't makes sense of it. Establish the mathematical objects rigorously, which you still have to do (unless I'm missing something obvious, which is likely as I am well known for being a useless mathematician, you ask my colleagues) and then demonstrate how it applies to the physical universe in exactly what ways. At the moment I can't correlate it with my own understanding of quantum mechanics which has been arrived at via the Schroedinger and Heisenberg methods. --prime mover 02:46, 19 January 2012 (EST)
 * As I am sure you know, $z$ is a positive integer if there exists a whole number $n \in \mathbb N^*$ such that $z = \{(j,k) \in \mathbb N^2: j = n +k\}$.  $z$ is a negative integer if there exists a whole number $n \in \mathbb N^*$ such that $z = \{(j,k) \in \mathbb N^2: n + j = k\}$ and $z = 0_{\mathbb Z}$ if $z = \{(j,k) \in \mathbb N^2: j = k\}$.  (This is why I call $\mathbb N^2$ the set of $proto$integers, since all integers are sets of protointegers.)  The set of non-negative integers is, of course, isomorphic to the ringoid of natural numbers.
 * I agree that if $(L_1,\vee_1,\wedge_1)$ and $(L_2,\vee_2,\wedge_2)$ are lattices then $(L_1 \times L_2, \vee, \wedge)$ is a lattice (with $\vee$ and $\wedge$ defined in the obvious way). This implies that $\mathbb N \times \mathbb N$ is a lattice, since $\mathbb N$ is a lattice. --KBlott 04:15, 19 January 2012 (EST)
 * Yes I know all that, but you still didn't answer what I asked: what is the ordering on NxN to make it a lattice? --prime mover 08:46, 19 January 2012 (EST)
 * It appears that if there are $n_1$ lattices on $L_1$ and $n_2$ lattices on $L_2$ then there are $n_1 \cdot n_2$ lattices on $L_1 \times L_2$.  So, since there are two lattices on $\mathbb N$, there should be four lattices on $\mathbb N \times \mathbb N$.  One of the four corresponding orderings is (obviously) consistent with the ordering on $\mathbb Z$.  It will take some time to confirm (or refute) this proposition. --KBlott 15:30, 21 January 2012 (EST)

Editing protocol
Please try and adhere to the following:

a) Please consider the use of the "This is a minor edit" checkbox for minor page edits. It then makes it easier to filter out trivial changes when reviewing recent changes.

b) Subpages are named using a forward slash / not a backslash \ otherwise the mediawwiki software does not understand that it is a page delimiter.

c) When you write a proof, please try and put it into the house style.

There's probably more, but that will do as a start. --prime mover 01:20, 18 January 2012 (EST)

Class structure WRT equivalence
Has the fact that these definitions (like Definition:Lattice) all use the terminology 'wrt $\approx$' to do with the fact that one cannot properly define a structure similar to $\mathcal C / \approx$ for sets? Though at first glance one can express the property of being an equivalence class wrt. $\approx$, which is then enough to define the class of equivalence classes, is it not? I am probably making some errors here; if you spot them, please point them out (and give the correct version). --Lord_Farin 02:59, 24 January 2012 (EST)
 * Regarding this diff. Notice that the diff adds a link from this page (which talks about equivalence relations on classes) to this page (which talks about equivalence relations on sets).  This should not be a problem if we want to talk about a class as small as a Statistical Ensemble of coordinate transformations on Minkowski space.  However, if we want to talk about the surreal numbers some modification will be necessary. --KBlott 11:13, 24 January 2012 (EST)


 * When classes are properly developed on PW, there are two courses of action, namely: 1. Change all pages to cover classes explicitly instead of sets. 2. Add a page for the corresponding definition for classes, referencing to the particular definition for sets it should reduce to.
 * That is, you are right; the change was made to comply with house style, and as for now, most stuff is for sets, not classes. Now could you please answer my questions? --Lord_Farin 11:21, 24 January 2012 (EST)
 * I don’t know if I can answer your question. Can you restate it? One well known example of $\approx$ is the set of integers, which is a partition on the set $\mathbb N^2$ = $\mathbb N \times \mathbb N$.  Two elements of  $\mathbb N^2$ are typically regarded as equivalent ($\approx$) if they belong to the same block in $\mathbb Z$.  Consider that algebra.  Explicitly, $(\mathbb N^2,+,-,0)$ where $+:\mathbb N^2 \times \mathbb N^2 \to \mathbb N^2$ is the binary operation such that for every $n =(n_0, n_1) \in \mathbb N^2$ and every $m =(m_0, m_1) \in \mathbb N^2$, $n + m := (n_0 + m_0, n_1 + m_1)$.  $-: \mathbb N^2 \to \mathbb N^2$ is a unary operation on $\mathbb N^2$ such that $\forall n \in \mathbb N^2$, $-n := (n_1, n_0)$ and $0$ is the nullary operation on $\mathbb N^2$ such that $0 := (0,0)$.  $(\mathbb N^2,+,-,0)$ is a monoid and a group wrt $\approx$.  However, it is not a group wrt $=$.  Therefore, when we are talking about an algebra which admits to a certain structure with respect to an equivalence relation that is not identity, we must state that relation explicitly or it will lead to problems.  I assume that this is the reason that these authors adopted the convention.
 * I certainly agree that stating $\approx: C \times C \to B$ explicitly and proving that it is an equivalence relation is sufficient to define the equivalence classes on $C$ wrt $\approx$ provided $B = \{\top, \bot \}$ with $\vee$, $\wedge$, and $not$ defined in the usual way. (In other words, I think it is reasonable to talk about $C/\approx$ in most cases provided we are using standard propositional calculus.)--KBlott 12:29, 24 January 2012 (EST)