On Wednesday, January 26, 2022 at 8:55:34 AM UTC-5, Timothy Golden wrote:
> On Tuesday, January 25, 2022 at 7:10:40 PM UTC-5, FredJeffries wrote:
> > On Tuesday, January 25, 2022 at 3:40:39 PM UTC-8, timba...@gmail.com wrote:
> > > On Tuesday, January 25, 2022 at 2:30:44 PM UTC-5, FredJeffries wrote:
> > > > On Tuesday, January 25, 2022 at 10:20:42 AM UTC-8, timba...@gmail.com wrote:
> > > > > On Monday, January 24, 2022 at 8:12:14 PM UTC-5, FredJeffries wrote:
> > > > > > On Monday, January 24, 2022 at 12:17:06 PM UTC-8, timba...@gmail.com wrote:
> > > > > > > On Sunday, January 23, 2022 at 3:15:19 PM UTC-5, Dan Christensen wrote:
> > > > > > > > On Sunday, January 23, 2022 at 11:41:47 AM UTC-5, Mostowski Collapse (aka Jan Burse) wrote:
> > > > > > > > > I guess there was also not a single mention of the number zero, right?
> > > > > > > > > That happens when you have a degree from the McDonalds university.
> > > > > > > > >
> > > > > > > > > But even if you ban the empty set based function space, your
> > > > > > > > > doing is nonsense. You will not be able to prove:
> > > > > > > > > > There is this famous theorem:
> > > > > > > > > > |A| = n -> |{ f | f : A -> 2 }| = 2^n
> > > > > > > > > Take |A| = 1. Your set { f | f : A -> 2 } is again the universal class.
> > > > > > > > Are you referring to the set of functions f: {x} --> {0, 1} for some x? If so, let's see your proof of how this leads to a universal set in DC Proof.
> > > > > > > > Dan
> > > > > > > >
> > > > > > > Isn't a function which returns the empty set the function that you are discussing?
> > > > > > No.
> > > > > >
> > > > > > The function IS the empty set. The empty set IS the function.
> > > > > >
> > > > > > The return type is pretty much irrelevant.
> > > > > >
> > > > > > The DOMAIN of the function is the empty set. So the parameter type is elements of the empty set, lets call it 'eotes' for short.
> > > > > >
> > > > > > So the function signature is
> > > > > >
> > > > > > eotes func(eotes x), or int func(eotes x), or ...
> > > > > >
> > > > > > If we memoize func to a table, we get an empty table -- just the headers, because there are no eotes.
> > > > > >
> > > > > > Since everything in the discussion is sets, functions are sets of ordered pairs with certain properties. Thus we have func memoized to a set of ordered pairs with no elements -- the empty set.
> > > > > Well you've completely ignored the sets here.
> > > > There is only one set relevant to the topic under discussion: The empty set. Yes, I DID mention it.
> > > > > I do believe that the set S at the very least would deserve consideration.
> > > > I have no idea to what your 'set S' refers. It doesn't matter.
> > > You gotta be kidding me. You are pulling a fluffer-nutter here?
> > Your rant has nothing to do with the subject under discussion: Empty functions (read the title of the thread)
> >
> > EOD
> > > f(x) is in S2. x is in S1.
> > > We could for instance offer up double precision IEEE values
> > > double from( int s )
> > > {
> > > double d = s;
> > > return d + 0.123456789;
> > > }
> > >
> > >
> > > from() : int -> double
> > > Two sets, and neither of them pondered by you.
> > > >
> > > > For ANY set S, there is exactly one function from the empty set to S: The empty function.
> > > False.
> > > Les S be the integers. Let f(NULL) be 2 in S. Let ff(NULL) be 3 in S. Let fff(NULL) be 1 in S
> > > There are at least three.
> > > Therefore there is not exactly one function from the empty set to S.
> > > >
> > > > If S is not the empty set, there are no functions from S to the empty set.
> > > I confirm that S is not the empty set. There are however plenty of functions from S to the empty set, but none of them are ones you've already been speaking of. You are introducing inverses for no apparent reason, but f', ff', and fff' as inverses do seem to exist. So here again I guess I am falsifying your own statement. You were speaking of empty set to S; not the other way around. Keeping it simple would be helpful. I'll do my best to carry on not becoming carrion.
> > > > > When you get to it and you introduce S into your description here doesn't the thing go kaboom?
> > > > Hasn't gone kaboom in 150 years.
> > > >
> > > > I'll repeat the references for 'initial object' and 'empty function'
> > > > > > https://en.wikipedia.org/wiki/Initial_and_terminal_objects
> > > > > > https://academickids.com/encyclopedia/index.php/Initial_object
> > > > > > http://nlab-pages.s3.us-east-2.amazonaws.com/nlab/show/initial+object
> > > > > > https://www.youtube.com/watch?v=yeQcmxM2e5I
> > > > > > https://math.stackexchange.com/questions/1439639/initial-and-final-objects-in-a-category
> > > > > > https://www.quora.com/Why-is-the-empty-set-an-initial-object-in-the-Category-Set
> > > > See also the textbook Hewitt & Stromberg's "Real and Abstract Analysis" where, on page 31, the empty function is crucially used in the proof that any two bases of a vector space have the same cardinality.
> > > It's kind of interesting to enter constants in as functions. It sort of fills out the operator theory from the instantiation angle. It is backwards looking, but it does seal the elemental format nicely. For all you bickering about S and empty sets of course constants answer the problem. And it is good. It takes on a wholesome feeling after all that fluffer-nutter you have us gag down.
> No Fred. You say that for any set S there is exactly one function from the empty set to S. I suppose the integers, and sure enough there are plenty of f(). The whole analysis of becoming reliant upon nothing within a theory is sort of a lame-ass position really. To broaden the scope let's ponder some other f() that could do something quite interesting:
> int f(){ static i = 0; return i++ }
> Now we have not exactly a constant function, but a consistent function none the less. Now the omniscience of the function is debunked. The idea that empty in means empty out somehow you've landed upon, yet that may be just one instance of a dark type more akin to P1 really in its zero dimensional state. If there is one thing we can give a number is it memory? The number which remembers itself? Then the function to act upon the number not as a thing more fundamental than an operator, but put correctly in its place atop the operators and ultimately as an operator. This sort of functional analysis I could get behind and let have a little action. She will run sweet all the time and expose the system as much as she hides her memories. I was once ten and then I turned eleven. twelve and twenty one were a decade of rapid change for all of us I would think. Yet which was the more coherent of us? Was it then or is it now and wouldn't we rather retain it for both the here and the now? That the now is the trick form and that the here need only be instantiated yet what the real and their ilk have done has landed us here. It is like a church run amuck. All of us its parish. Perish we mustn't. Squeak We Must!@

If we allow in this function with a numerical memory; no different than a constant yet operant as well as in the sense of the function... well, the sky is the limit and the freedom to construct is extreme. This is as it should be. This is what we need. We need rich data from naught. To get physical reality we need rich data from naught. As to what selection criteria we require: at this primitive level we need none. We need a scaffol and it is good that the scaffold not be confused with the painter, though of course the painter holds the scaffolding up, and by this a cryptic language ensues yet the pursuit of mathematics is to do so in an unencrypted form. So we go carefully. We go slowly. Yet in a place where falsifications are not addressed and the freedom to construct is so large, and what? You are going to have a problem with a number with a memory? Hah! Now we have a true parishioner..

On Saturday, January 22, 2022 at 7:33:07 PM UTC-5, Dan Christensen wrote:
> Are empty functions (those with empty domains) important in real analysis or algebra? I wouldn't think so.
>

This turns out to be a non-issue, in DC Proof. The Function Axiom is simply not applicable for empty domains or codomains. It introduces the standard "f(x)" functional notation, but only for non-empty domains and codomain. Wouldn't make sense otherwise.

Dan

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