Naive about Set Theory

A few months back I started putting together a page about systems to provide some foundations (framing, schema?) to communicate further expressions about value-systems, their nature, and evaluating their efficacy and ultimately “value” across a variety of value-assessing criteria (and as one might guess, explain criteria for evaluating any criteria making up an identifiable value system). In other words, what is the value of having a value-system, and how can value-systems be evaluated…and why they must be…

But that’s not directly germane to my topic title here. Instead, while trying to articulate the definition of a system (using the definition of a thermodynamic system as a starting point) I came up to the point describing systems as having boundaries. I found myself started trying to “justify” the definition, and explaining why systems had to have boundaries.

None of what I was writing seemed to click with me. I couldn’t put my finger on it, but felt I was dancing around something. Eventually for reasons that I find murky now, I started down a path that led me to read about set theory (I think I was trying to get to a point where I could read John von Neumann), and ultimately naive set theory (which is a predecessor or sorts to NBG set theory). While reading about set theory, I think I came across what I was having a problem with: a strong understanding about the nature and use of axioms in conceptual structure building.

Generally, axioms are those things in a theoretical structure that are taken as a given; effectively a truth onto which things must get anchored to have any hope of being considered truthful in the structure of the field of study under question.

What I needed to go forward with my writing about “systems” was to understand, assert (even if only implicitly to myself) the axiomatic nature of the terms I was putting forth, and move on from there. I found that slightly unrewarding, as like many people I seek to “get to the bottom” or ultimate justifiability of a position I take. Axioms are like bedrock, but I want to know what makes up the bedrock. Why is it bedrock? That’s for another entry also: the Axiomatic Axioms.

Why I was having a problem was because axioms are effectively the boundaries of the “systems” of theoretical topics; they have to be accepted (as valuable) within the topic area. There is no reason a person could state that a system doesn’t need to have a boundary, but then the system would be “everything”. A large part of my interest on systems is why and where the boundaries are drawn is largely dependent on the value-systems that identify or construct the systems.

Anyway, I’m sort of reading through Mary Tiles book on the Philosophy of Set Theory, mainly because Naive Set Theory by Paul R. Hamos was starting to get away from me. I could sort of follow it, but it was written to support an academic classroom mindset, so it presents thing and expects the instructor to assist in the understanding. This means it packs concepts in pretty quick and high if you just read the book.

The Philosophy of Set Theory is a bit easier for me as it delves more into the epistemological history of the considerations of the various infinities and infinitesimals that led to modern set theory. It also has some diagrams.

Author: sageikosa

Ikosa Framework Author

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