A question about the empty set

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Discussion Overview

The discussion revolves around the concept of the empty set within set theory, particularly in the context of Zermelo-Fraenkel (ZF) axioms. Participants explore foundational ideas about sets, the nature of existence in set theory, and the implications of hidden assumptions within axiomatic frameworks.

Discussion Character

  • Exploratory
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • Some participants argue that a clear definition of sets is crucial, emphasizing the distinction between container and content.
  • Others propose that the ZF axiom of the empty set contains hidden assumptions about the nature of content, suggesting that if content is "nothing," then the empty set cannot be defined as such.
  • A participant draws an analogy to probability theory, questioning whether axioms can exist without defining their terms.
  • Some participants challenge the notion that there are hidden assumptions in ZF, asserting that the existence of sets and elements is fundamental to the theory.
  • There is a discussion about the implications of distinguishing between an "x-model" and "x itself," with some arguing that this distinction is not necessary in ZF.
  • One participant suggests that basic concepts in set theory, like membership and the definition of sets, are taken as self-evident, similar to primitive concepts in geometry.

Areas of Agreement / Disagreement

Participants express differing views on whether the ZF axioms contain hidden assumptions and whether the concept of the empty set is adequately defined. There is no consensus on these points, and the discussion remains unresolved.

Contextual Notes

Participants note that the discussion involves complex interpretations of mathematical notation and conventions, with some expressing confusion over terms like "x-model" and the implications of "hidden assumptions." The debate highlights the nuances in foundational concepts of set theory.

  • #31
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  • #32
erm, at whihi point did you define locality?

as for the other post. you adequately answered my question about entropy - a number is entropic iff it is composite - there is a non-trivial partition of n into r equal parts. 1<r<n.

ok, i;ll go over there and write more do not answer to that in this thread
 
  • #33
Hurkyl wrote:

By "set's content", are you referring to the text that is between '{' and '}' when we write a set in that notation?

Organic replied:

A text is nothing but a representation of some idea.

please look at my previous post, thank you.

The words "yes" and "no" are typically very useful words for responding to questions.
 
  • #35
Dear Hurkyl,

No.
 
  • #36
Organic, again, wtf are locality and non-locality?
 
  • #37
Matt,

Please look at the link in my previous post, thank you.
 
  • #38
the link does not contain any reference to the locaity. moreoever all the consrtuctions there are self contradictory as you use the empty set, which you mantain makes no sense.

in fact, and i reserve the right to correct myself later, it appears that pdf is almost entirely complete and utter bull****. non-technical term.

you say clearly there are | and _ structures. sorry, but 'clear' in whose deluded mind?

and btw you are misusing the term 'base'.

and you are misusing the * in Z*. * usually means the multiplicative group, which in Z would be {1,-1}

but apart from those gross errors the rest is just specious drivel.

and yes, you've exhausted my patience.
 
  • #39
Matt,

Have a nice run after your scholastic tail.
 
  • #40
so, if i get this right, you are allowed to use whatever definition you want for some object/concept, without saying what that object/concept is, and then, when someone says you are wrong to do so, you throw a tantrum about them not seeing the correct bigger picture, and that they are 'scholastic'? why on Earth post your ideas in math forum if you aren't going to adhere to the basics of mathematics? you say something exhibits property of locality but won't say what locality is! is it unreasonalbe to ask you to explain that? apparently.
 
  • #41
I'm going to leave the bit about logic aside for a moment.


Let's start with some basics.

ZF gives us an axiomatic definition of a set.

Is what you call a set the same as what ZF calls a set?



There exists what I would call a set that is denoted by the string of symbols

{5, q, apple, unicorn, gazorninplat}

.

Does this string of symbols denote what you would call a set? What is the content of this set, by your meaning?
 
  • #42
Ah, c'mon matt, you are starting to sound like me!

I thought I had the "grumpy old man" concession on this forum!

(You may have noticed that I have stopped making any response to Organic whatsoever. Blood pressure problem.)
 
  • #43
Dear Hurkyl,

It is a very good question.

Answer:

Code: 
{5, q, apple, unicorn, gazorninplat}
 |  |   |       |         |  
 |__|   |       |         | 
 |      |       |         |
 |______|       |         |
 |              |         |
 |______________|         |
 |                        |
 |________________________|
 |
 |
Or more general:
Code: 
{., .,  .,      .,        .}
 |  |   |       |         |  
 |__|   |       |         | 
 |      |       |         |
 |______|       |         |
 |              |         |
 |______________|         |
 |                        |
 |________________________|
 |
 |

If any '.' in this set is some unique object with no common property to the other
objects (accept 'uniqueness'), then we can change the order of '.' in the set without changing the information structure, for example:
Code: 
{q, 5, unicorn, apple, gazorninplat}
 |  |   |       |         |  
 |__|   |       |         | 
 |      |       |         |
 |______|       |         |
 |              |         |
 |______________|         |
 |                        |
 |________________________|
 |
 |

If '.' can be ordered by some common property we get:
Code: 
{q, 5, unicorn, apple, gazorninplat}
 |  |    |       |           |  
 |__|    |       |           | 
    |    |       |           |  
    |____________|           |
    |    |                   |
    |____|                   |
    |                        |
    |________________________|
    |
    |
If there are identical objects in some non-ordered collection, we can get for example:
Code: 
{5, 5,  q,   unicorn, apple, gazorninplat}
 |  |   |       |         |     |  
 |__|_  |       |         |     | 
 |      |       |         |     |
 |______|       |         |     |
 |              |         |     |
 |______________|         |     |
 |                        |     |
 |________________________|     |
 |                              |   
 |______________________________|
 |
 |
And if there is a common order we have:
Code: 
 q  q 
{5, 5, apple, unicorn, gazorninplat}
 |  |   |       |         |  
 |__|_  |       |         | 
 |      |       |         |
 |______|       |         |
 |              |         |
 |______________|         |
 |                        |
 |________________________|
 |
 |
 
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  • #44
Originally posted by HallsofIvy
Ah, c'mon matt, you are starting to sound like me!

I thought I had the "grumpy old man" concession on this forum!

(You may have noticed that I have stopped making any response to Organic whatsoever. Blood pressure problem.)

I know, yet another young idealist comes up against a pig-headed refusal to learn and eventually cracks under the stress. It's why I should never become a teacher.

This is a temporary thing I hope - I will go back to simply pointing things out without losing it. The new tactic will be one objection at a time so Organic can't wander off topic as he tends to.
 
  • #45
It is a very good question.

Answer:...

My best guess is that you're trying to demonstrate content, via the pictures like

Code: 
{5, q, apple, unicorn, gazorninplat}
 |  |   |       |         |  
 |__|   |       |         | 
 |      |       |         |
 |______|       |         |
 |              |         |
 |______________|         |
 |                        |
 |________________________|
 |
 |

Well, what is content? Is it this entire picture? Is it just

Code: 
 |  |   |       |         |  
 |__|   |       |         | 
 |      |       |         |
 |______|       |         |
 |              |         |
 |______________|         |
 |                        |
 |________________________|
 |
 |

?

Is it

Code: 
5, q, apple, unicorn, gazorninplat

?

Is it actually one of these, or simply denoted by one of these?


What is '.'? It looks as if you're trying to use the five '.' as logical variables. Is this correct? Why not just say something like:

----------------------------------------------
Or more general:

Code: 
{a, b,  c,      d,        e}
 |  |   |       |         |  
 |__|   |       |         | 
 |      |       |         |
 |______|       |         |
 |              |         |
 |______________|         |
 |                        |
 |________________________|
 |
 |

where a, b, c, d, and e are any 5 distinct objects
----------------------------------------------


I would still like an answer to my other questions:

ZF gives us an axiomatic definition of a set.

Is what you call a set the same as what ZF calls a set?



There exists what I would call a set that is denoted by the string of symbols

{5, q, apple, unicorn, gazorninplat}

.

Does this string of symbols denote what you would call a set?
 
  • #46
As I wrote, for me a set's content is the model of some x, where '{' and '}' is the framework, which we use to examine x-model.

1) x is (model of nothing)
2) x is . (model of locality)
3) x is __ (model of non-locality)

a,b,c,... is like 5, q, apple,... but I go simpler than that and use '.' as general notation for some singleton, and '_' for general notation for non-singleton.

To understand singleton, non-singleton please read:

http://www.geocities.com/complementarytheory/P0is1.pdf
 
  • #47
so __ is your notation for some 'collection' of elements (note the plural). You also use it for some limit of everything elsewhere (in aspirating I seem to recall; still not changed the name, eh?), and you say that |{_}| =1 so how can a set with more than one element have cardinality 1? no wonder you're struggling to understand this stuff.

what is x?
 
  • #48
No Matt,

__ is not a collection and also . . . is not a collection.

Code: 
Only the association between _ and . is a collection, 
for exemple:

{.,.,.}
 | | | 
 |_|_|_
 |

'{' and '}' is the framework, which we use to explore these collections.

x is any thing that can be thinkable by us.
 
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  • #49
I was attempting to say, and I think I implied it, that . is a single element of some 'set' and __ denotes contents of some non-singleton 'set' ie the collection of elements in the 'set'. Is that how we're supposed to interpret your statement?


I now also choose x to be all those things that do not have a model as contents for some set in organic's theory. that is thinkable. i just thought it.

that seems to be about a problem there doesn't it? seeing as a set's contents is some x-model
 
  • #51
so it is a model that isn't a model?
 
  • #52
x content cannot be but a model of the content.

No content has any impact on this hierarchy.
 
  • #53
Let x1 = the model of "all not x-models"

Let x2 = the model of "the model of "all not x-models""

x1 not= x2

general notation:

x=model(x)


x = x only if x is actual(= not theoretical, or not a model)
 
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  • #54
x=x is a tautology.

you know, you might want to define what you mean by model at some point, as even you seem confused by it.
 
  • #55
I am not confused, you are the one because your point of view does not distinguish between the theoretical and the actual.
 
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