# Foundation Gaps: Natural Numbers, Expressions, Sets, and Finite Limit Models

Deepen cross-domain question families on seven existing knowledge objects and their prerequisites, keeping mathematical sources unchanged and checking boundaries in exact finite slices.

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This is a suggested learning sequence, not a universal logical dependency or a formal proof.


- Stage 1 · optional



## [Natural number](/en/mko/mko-natural-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261011%2F%3Ffrom%3D%252Fen%252Fpaths%252F)

Fix N including 0; list successor windows, two partial predecessors, closed-interval sizes, and rational candidates for N. Zero has no natural predecessor in this protocol; do not conflate it with truncated pred, negative values, or fractions.

[Related questions](/en/explore/?mko=mko-natural-number) [Concept guides](/en/derivations/?q=mko-natural-number)


- Stage 1 · optional



## [Set](/en/mko/mko-set/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261011%2F%3Ffrom%3D%252Fen%252Fpaths%252F)

Use finite sets of an explicit integer type: deduplication/sorting, equality with both differences, candidate membership partitions, and complete powersets of at most four elements. The empty set’s powerset contains the empty set; no unrestricted comprehension or new axiom is asserted.

[Related questions](/en/explore/?mko=mko-set) [Concept guides](/en/derivations/?q=mko-set)


- Stage 1 · required



## [Set membership](/en/mko/mko-set-membership/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261011%2F%3Ffrom%3D%252Fen%252Fpaths%252F)

Existing canonical prerequisite for this density pulse.

[Related questions](/en/explore/?mko=mko-set-membership) [Concept guides](/en/derivations/?q=mko-set-membership)


- Stage 1 · required



## [Function mapping](/en/mko/mko-function-mapping/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261011%2F%3Ffrom%3D%252Fen%252Fpaths%252F)

Existing canonical prerequisite for this density pulse.

[Related questions](/en/explore/?mko=mko-function-mapping) [Concept guides](/en/derivations/?q=mko-function-mapping)


- Stage 2 · optional



## [Arithmetic operations](/en/mko/mko-arithmetic-operations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261011%2F%3Ffrom%3D%252Fen%252Fpaths%252F)

Use rational arithmetic, or natural inputs with Q results: ordered four operations, parenthesized pairs, inverse candidates with signed defects, and natural eligibility of differences/quotients. Division by zero is undefined, negative differences are not truncated to zero, and quotients are not integer division.

[Related questions](/en/explore/?mko=mko-arithmetic-operations) [Concept guides](/en/derivations/?q=mko-arithmetic-operations)


- Stage 2 · optional



## [Subset](/en/mko/mko-subset/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261011%2F%3Ffrom%3D%252Fen%252Fpaths%252F)

Compute all A-outside-B obstructions, non-strict/strict inclusion with both witnesses, three chain-obstruction sets, and typed candidate masks. Empty-set inclusion and equality are not strict inclusion, and finite checks are not universal set proofs.

[Related questions](/en/explore/?mko=mko-subset) [Concept guides](/en/derivations/?q=mko-subset)


- Stage 2 · optional



## [Sample space](/en/mko/mko-sample-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261011%2F%3Ffrom%3D%252Fen%252Fpaths%252F)

Use explicitly declared possible-outcome protocols: ordered products, recording images and original fibers, missing/outside candidate outcomes, and separate outcome/event masks. Assign no probabilities and infer neither uniformity, independence, nor completeness of an unknown experiment.

[Related questions](/en/explore/?mko=mko-sample-space) [Concept guides](/en/derivations/?q=mko-sample-space)


- Stage 2 · optional



## [Tends-to relation](/en/mko/mko-tends-to-relation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261011%2F%3Ffrom%3D%252Fen%252Fpaths%252F)

Both filters are explicit finite principal filters, with a total map on complete finite types. Check image generators, f(S)-outside-T obstructions, typed candidates, and two composition-image routes; empty generators and the bottom filter are allowed. This finite model does not interpret a bare arrow as a general sequence/point limit or create new formal Evidence.

[Related questions](/en/explore/?mko=mko-tends-to-relation) [Concept guides](/en/derivations/?q=mko-tends-to-relation)


- Stage 3 · optional



## [Variables and algebraic expressions](/en/mko/mko-variable-expression/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261011%2F%3Ffrom%3D%252Fen%252Fpaths%252F)

Specify four small expressions over Q, not a universal AST: two environments, simultaneous variable exchange, lexical shadowing, and the original domain of a/(x−b). A zero numerator does not remove an excluded point; expressions are not equations.

[Related questions](/en/explore/?mko=mko-variable-expression) [Concept guides](/en/derivations/?q=mko-variable-expression)



Path source & statuspath-pulse-foundations-20261011 · candidate

[Original paths JSON](/data/architecture/learning-paths.json)