Sets being subsets of themselves

[guitarplayer]

In what way could we imagine a number of sets being all each other's subsets? @daylen maybe?

This is inspired by Goedel, I'm reading about Boyd's work.

[ducknald_don]

Russell's Paradox:

https://en.wikipedia.org/wiki/Russell's_paradox

[7Wannabe5]

Goedel has to do with self-referential statements and boundaries.

Maybe think* about when you pull the last garbage bag out of the box that contains the garbage bags and then you throw away that box into the bag.

*Don't do, recycle! lol

[daylen]

From https://en.wikipedia.org/wiki/Self-reference we have..

In mathematics and computability theory, self-reference (also known as impredicativity) is the key concept in proving limitations of many systems. Gödel's theorem uses it to show that no formal consistent system of mathematics can ever contain all possible mathematical truths, because it cannot prove some truths about its own structure. The halting problem equivalent, in computation theory, shows that there is always some task that a computer cannot perform, namely reasoning about itself. These proofs relate to a long tradition of mathematical paradoxes such as Russell's paradox and Berry's paradox, and ultimately to classical philosophical paradoxes.

In game theory, undefined behaviors can occur where two players must model each other's mental states and behaviors, leading to infinite regress.

In computer programming, self-reference occurs in reflection, where a program can read or modify its own instructions like any other data.[3] Numerous programming languages support reflection to some extent with varying degrees of expressiveness. Additionally, self-reference is seen in recursion (related to the mathematical recurrence relation) in functional programming, where a code structure refers back to itself during computation.[4] 'Taming' self-reference from potentially paradoxical concepts into well-behaved recursions has been one of the great successes of computer science, and is now used routinely in, for example, writing compilers using the 'meta-language' ML. Using a compiler to compile itself is known as bootstrapping. Self-modifying code is possible to write (programs which operate on themselves), both with assembler and with functional languages such as Lisp, but is generally discouraged in real-world programming. Computing hardware makes fundamental use of self-reference in flip-flops, the basic units of digital memory, which convert potentially paradoxical logical self-relations into memory by expanding their terms over time. Thinking in terms of self-reference is a pervasive part of programmer culture, with many programs and acronyms named self-referentially as a form of humor, such as GNU ('GNU's not Unix') and PINE ('Pine is not Elm'). The GNU Hurd is named for a pair of mutually self-referential acronyms.

Tupper's self-referential formula is a mathematical curiosity which plots an image of its own formula.

This is one of those themes that opens up a liminal intellectual space between the disciplines of study. Everything is infinitely inter-connected, and some spiders find themselves crawling onto super-connective webs that started being weaved eons ago.

[daylen]

In the domain of social science and technology, I like to associate self-reflection with "social mirrors" and self-reference with "data washing machines". Artificial intelligence helping to interpolate styles between cultural icons and to extrapolate novel styles from icon deconstruction and reconstruction. Allowing more and more agents to peer into the collective consciousness and outline collective shadows based upon their own projections. Dogmas and prophecies will start to spawn and grow from recursive identification and instrumentalization, leading to an averaging of recorded culture (that is trained upon).. a religion of the future that is unlike any religion of the past yet grounded in them. Ironically, this homogenization will afford more local or regional diversity that in turn may sustain creative explosion.

[ducknald_don]

A Quine is a program that print its own source:

https://en.wikipedia.org/wiki/Quine_(computing)

[guitarplayer]

Good! Thank you, I wished I could say more but I can't just now. Russell knocking at my door again. @7w5, very imaginative as always. @daylen a bunch of keywords to explore; also, nice to read a definition of bootstrapping I get.

[BeyondtheWrap]

In what way could we imagine a number of sets being all each other's subsets?

If A is a subset of B, and B is a subset of A, then A = B.

Do you just mean a longer version of this with more than 2 sets? Say, n sets where each set is a subset of the next, and the last is a subset of the first to close the loop? I’m pretty sure the same result would happen; they all would have to be identical.

[guitarplayer]

I meant a situation when A =/= B.

Initially my thinking was along the lines of e.g.:
- physicists talk about atoms. 'Atoms' is a subset of 'physicist'
- atoms are what makes physicists. 'Physicist' is a subset of 'atoms',

or

- anthropological/cognitive analysis of hard sciences (Bruno Latour 'Science in Action'; theories of cognition; epistemology and philosophy of science) - hard sciences being a subset of all the stuff that can be anthropologically/cognitively analyzed
- hard scientific analysis of social science/anthropology/cognitive psychology/mental models (reductionism, psychophysical parallelism) - the latter being a subset of all the things that can be analyzed with hard sciences.

I am not entirely sure I am asking the right question here.

[daylen]

This line of thinking runs into Russell's paradox in such a way that can be thought of as a type error. A physicist is a student of physics which refers to atoms (physics talks about atoms and is thus not of atoms), thus a physicist is of a different logical level than an atom. This can be seen more clearly in type theoretic foundations of comp sci and math where a strict typing chain is necessary to avoid nonsensical algorithms or halts.

Also relates to the picture of the hand writing itself into existence. Or to how "God" must be both a constructor and its own instantiation (i.e. what we can "see" is not what causes us to see what we see, but somehow "God" reconciles this dichotomy). Also see constructor theory and the hard problem of consciousness.

The book Godel, Escher, Bash is helpful here. Generally, parallels can be drawn between formal systems of language, impossible drawings or processes, harmonics of the senses (musical symmetry), and so forth.

[guitarplayer]

This book

https://en.wikipedia.org/wiki/G%C3%B6de ... er%2C_Bach

Wow, its description is right down my alley, thanks!

People on this forum give such good references.

[Tyler9000]

The concept reminds me of the proposal that every subatomic particle contains a full universe of its own, and our own known universe lies within one particle of something even larger. Even if the whole system is infinitely large and infinitely small without looping back on itself, the entire concept of universe and atom does get much harder to define.

[guitarplayer]

Roger Penrose here talks about the infinity, time and the universe. The far-future-universe-forgetting-about-itself being the starting point of itself.

[daylen]

Interesting tangent into mereology, from https://en.wikipedia.org/wiki/Mereology:

Reflexivity: A basic choice in defining a mereological system, is whether to consider things to be parts of themselves. In naive set theory a similar question arises: whether a set is to be considered a "subset" of itself. In both cases, "yes" gives rise to paradoxes analogous to Russell's paradox: Let there be an object O such that every object that is not a proper part of itself is a proper part of O. Is O a proper part of itself? No, because no object is a proper part of itself; and yes, because it meets the specified requirement for inclusion as a proper part of O. In set theory, a set is often termed an improper subset of itself. Given such paradoxes, mereology requires an axiomatic formulation.

A mereological "system" is a first-order theory (with identity) whose universe of discourse consists of wholes and their respective parts, collectively called objects. Mereology is a collection of nested and non-nested axiomatic systems, not unlike the case with modal logic.

The treatment, terminology, and hierarchical organization below follow Casati and Varzi (1999: Ch. 3) closely. For a more recent treatment, correcting certain misconceptions, see Hovda (2008). Lower-case letters denote variables ranging over objects. Following each symbolic axiom or definition is the number of the corresponding formula in Casati and Varzi, written in bold.

A mereological system requires at least one primitive binary relation (dyadic predicate). The most conventional choice for such a relation is parthood (also called "inclusion"), "x is a part of y", written Pxy. Nearly all systems require that parthood partially order the universe. The following defined relations, required for the axioms below, follow immediately from parthood alone:

An immediate defined predicate is "x is a proper part of y", written PPxy, which holds (i.e., is satisfied, comes out true) if Pxy is true and Pyx is false. Compared to parthood (which is a partial order), ProperPart is a strict partial order.

... a bunch of typeset axioms ...

M8 is also called "General Sum Principle", "Unrestricted Mereological Composition", or "Universalism". M8 corresponds to the principle of unrestricted comprehension of naive set theory, which gives rise to Russell's paradox. There is no mereological counterpart to this paradox simply because parthood, unlike set membership, is reflexive.

[7Wannabe5]

How about situations such as negative integers, positive integers and the integer zero?

[daylen]

I assume you meant to respond to viewtopic.php?p=261990#p261990 ?

Not an expert but it seems that to demonstrate an incompleteness proof, statements made by an axiomatic system must be counted (with natural numbers 1, 2, 3..). The natural numbers can be extended to the integers (..3, 2, 1, 0, 1, 2, 3..) by introducing a subtraction operator. Thus any system that can conceive of the integers can also count and is thus subject to the incompleteness theorems.

[7Wannabe5]

No, I was thinking about how I have to address the question “Is zero a negative or a positive number?” with my students and how much they don’t like my answer.

[daylen]

I would guess that is an indeterminate truth from within ZFC. That would mean zero cannot be said to be not positive, not negative, positive or negative, positive and negative.

Though, it is often the case that learning the wrong way first adds traction on the path towards the right way (which is unobtainable as an absolute formal system to rule them all).

For advanced high schools students and above(*) I would posit zero being both and zero being neither, while showing that zero being either/or requires an asymmetric treatment (are the natural numbers + 0 more real? Is -2 an imitation of 2?).

Below that, pluralistic truth may be more confusing than helpful, so I would stick with one of the interpretations depending on my mood.

Curious what you say?

(*) if I had the time and energy

[7Wannabe5]

Gotcha. My students are age 6 to 22 with median 8th grade level, so I generally stick with “neither.” The topic is further complicated by the fact that humans have a kinesthetic relationship with positive integers not inclusive of the possibility of zero AKA natural numbers, so the conceptual boundary is rather cliff-like. If I had a penny for every time I’ve said “Imagine you have dug a hole ...”

I think this is kind of relevant to the argument for reading vs doing or wissen vs kennen. Sometimes if you can manipulate the abstractions in your mind, you don’t have to go out in the world and dig a hole.

[jacob]

This reminds me of the extension to "the map is not the territory": specifically that "the territory is not the map either or is it?!?"

In particular, if your map of the territory has a sufficiently high resolution, you should see your map of the territory on your map of the territory. Zooming in further, in that map there will be another map. And so on all the way down. Maps within maps within maps... and eventually: what is really territory and what is really map? Wooooo...

The difference between practice and theory is how the former stops at the first level, whereas the latter pursues it all the way down w/o returning to the surface?

It seems isomorphic to the difference between thinking of infinity as "amazingly enormous everything" and abstracting it to "if for every N, there's always N+1, then the set of 'N' is infinite", workable, done!