Well Ordering Proof Calculator

Calculator inputs

The page stays single-column overall. The input controls use three columns on large screens, two on medium screens, and one on mobile screens.

Analysis mode

Finite set integers

Finite set chosen lower bound

Progression first term a

Progression common difference d

Progression sample term count

Progression lower bound filter

Polynomial coefficient A

Polynomial coefficient B

Polynomial coefficient C

Predicate relation

Predicate threshold

Theorem range start

Theorem range end

Example data table

Mode	Example input	Least element or failure	Interpretation
Finite set	{12, 5, 19, 7, 4}	Least element = 4	A nonempty finite set always has a smallest value.
Progression	a = 5, d = 3, lower bound = 20	Least qualifying term = 20	The first term meeting the bound anchors the proof.
Theorem explorer	f(n) = n² - 3n + 2, test f(n) ≥ 10	Least counterexample found by scan	The first failure becomes the candidate minimal counterexample.
Negative progression	a = 20, d = -2, lower bound = 0	Infinite set not well-ordered here	Descending terms warn that the proof setup needs revision.

Formula used

Well-ordering principle: Every nonempty subset of the natural numbers has a least element.

Counterexample set: \( C = \{ n \in \mathbb{N} : P(n)\text{ is false} \} \). If \( C \neq \varnothing \), then \( C \) has a least element \( m \).

Least element in a finite set: \( m = \min(S) \).

Arithmetic progression subset: \( S = \{ a + kd : k \in \mathbb{N}_0,\ a + kd \ge L \} \).

First qualifying progression index for positive d: \( k_0 = \max(0,\lceil(L-a)/d\rceil) \).

Least qualifying progression term: \( m = a + k_0d \).

Polynomial test expression: \( f(n) = An^2 + Bn + C \).

Proof strategy: Assume a counterexample exists, pick the least one by well-ordering, then contradict its minimality.

How to use this calculator

Choose the analysis mode that matches your proof idea.
Enter a finite set, a progression, or a polynomial statement.
Press the submit button to compute the least element or least failure.
Read the result block shown above the form and below the header.
Use the proof outline as a draft for class notes, homework, or verification.
Review the graph to see ordering, growth, or the first failing index.
Download the CSV for records or the PDF for sharing.
Refine inputs until the structure matches the theorem you want to prove.

Frequently asked questions

1. What does this calculator actually prove?

It does not replace a formal proof. It organizes sets, identifies least elements, finds minimal counterexamples, and drafts reasoning steps you can turn into a rigorous argument.

2. Why is the least counterexample important?

A least counterexample is the core contradiction tool. If a smaller case would force the statement at that least failure, the assumed counterexample cannot exist.

3. Can I use negative integers?

Yes. The finite set mode accepts negative integers. For infinite arguments, you usually need a subset translated or constrained so the proof fits a bounded-below framework.

4. Why does a negative common difference raise a warning?

A negative difference creates a descending infinite progression. That setup may fail the bounded-below requirement needed for a clean well-ordering argument.

5. Does checking a range prove a theorem?

No. Range checking gives evidence, not a universal proof. It helps you locate a likely least counterexample or build intuition before writing the symbolic argument.

6. When should I use finite set mode?

Use it when you want to verify ordering quickly, confirm a least element, or illustrate how a nonempty finite integer set satisfies the least-element idea.

7. What is the graph showing?

The graph displays sorted set values, sampled progression terms, or polynomial outputs across n. The horizontal reference line marks the chosen lower bound or threshold.

8. Can this support classroom examples?

Yes. The tool is useful for lecture demos, worksheet preparation, student practice, and checking whether a proposed minimal-counterexample setup is sensible.