The concept of a container is tangible, not abstract.

All boundary forms can be constructed in physical space.

A container is an Object from the Outside and a Process from the Inside. (Bricken, Iconic Arithmetic Volume I, p. 137)

We are representing physical containers by delimiting Boundaries and basic Facts by empty containers.

Containers are fundamentally different than Symbols, containers have a physical presence. Containers also have an inside. Here’s Lakoff and Núñez as they develop a theory about the cognitive origins of mathematics:

The Container schema has three parts: an Interior, a Boundary, and an Exterior. This structure forms a gestalt, in the sense that the parts make no sense without the whole. [⇒ Whole Makes Its Parts]

Containers represent Distinctions.

Distinction is perfect continence. (Laws of Form)

a single concept system that of containment

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LAKOFF, George and NÚÑEZ, Rafael E., 2000. Where Mathematics Comes From: how the embodied mind brings mathematics into being. New York, NY: Basic Books. ISBN 978-0-465-03770-4. pdf

The point here is that, within formal mathematics, where all mathematical concepts are mapped onto set-theoretical structures, the “sets” used in these structures are not technically conceptualized as Container schemas. They do not have Container schema structure with an interior, boundary, and exterior. Indeed, within formal mathematics, there are no concepts at all, and hence sets are not conceptualized as anything in particular. They are undefined entities whose only constraints are that they must “fit” the axioms. For formal logicians and model theorists, sets are those entities that fit the axioms and are used in the modeling of other branches of mathematics.

Of course, most of us do conceptualize sets in terms of Container schemas, and that is perfectly consistent with the axioms given above. However, when we conceptualize sets as Container schemas, a constraint follows automatically: Sets cannot be members of themselves, since containers cannot be inside themselves. Strictly speaking, this constraint does not follow from the axioms but from our metaphorical understanding of sets in terms of containers. The axioms do not rule out sets that contain themselves. However, an extra axiom was proposed by von Neumann that does rule out this possibility.

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