Calculator Form
Formula Used
Russell rule: R = { x | x ∉ x }
Membership test: x ∈ R ⇔ x ∉ x
Paradox substitution: R ∈ R ⇔ R ∉ R
The calculator checks whether the selected object can be safely tested. In naive mode, substituting R into its own definition creates contradiction. In typed or restricted modes, the dangerous self reference is blocked.
How To Use This Calculator
- Select the theory mode you want to test.
- Choose Russell set R or another object.
- Set whether the object contains itself.
- Add a membership assertion if you want to test consistency.
- Press the calculate button.
- Read the result above the form.
- Use CSV or PDF export for records.
Example Data Table
| Mode | Object | Self Membership | Derived Result | Status |
|---|---|---|---|---|
| Naive | Russell set R | Unknown | R ∈ R ⇔ R ∉ R | Contradiction |
| Naive | Ordinary set A | No | A ∈ R | Stable |
| Naive | Ordinary set A | Yes | A ∉ R | Stable |
| Typed | Russell set R | Any | Self membership rejected | Blocked |
| Restricted | Russell class | Any | Not admitted as ordinary set | Blocked |
Understanding Russell Paradox
Russell paradox is a famous problem in set theory. It appears when a rule creates the set of all sets that do not contain themselves. Call that set R. The question then becomes simple. Does R contain itself?
When R contains itself, it breaks its own rule. The rule says R should contain only things that do not contain themselves. So R should not contain itself. When R does not contain itself, the rule includes it. So R should contain itself. Both choices lead to the opposite result.
This calculator turns that idea into a clear logic test. It lets you choose a theory mode, an object, and a self membership claim. It then checks whether the claim is valid, contradictory, blocked, or incomplete. The output is educational, not a theorem prover. It explains the reasoning in plain steps.
In naive set theory, any defining property can form a set. That freedom creates the paradox. The tool models this with the rule R equals all x where x is not a member of x. When x is R, the result becomes R is a member of R if and only if R is not a member of R.
Typed set theory avoids this by separating objects into levels. A set cannot be a member of itself, because membership must move from one level to another. The calculator can show that the expression is blocked before contradiction appears. This is useful for comparing foundations.
The score panel gives a severity level. Low means the selected claim is stable. Medium means the result depends on unknown membership. High means a direct contradiction appears. Blocked means the theory disallows the expression.
A careful result prevents a common mistake. The paradox is not about ordinary collections like books or numbers. It is about unrestricted self reference. Seeing the rule line by line makes the conflict easier to understand.
Use this page for study notes, classroom examples, and quick demonstrations. Enter different objects, change assumptions, and compare outputs. Export the result when you need records for homework or a lesson handout. The example table also shows common cases. It helps learners see why the paradox matters and why modern foundations add restrictions.
FAQs
What does this calculator test?
It tests the Russell rule, self membership, and contradiction status. It helps learners see why unrestricted set formation creates a logical problem.
Is this a formal theorem prover?
No. It is an educational logic tool. It explains the paradox with simplified set theory choices and readable reasoning steps.
Why does R cause contradiction?
R is defined as the set of things not containing themselves. Testing R against that rule makes R contain itself only when it does not.
What does naive mode mean?
Naive mode allows any property to define a set. That broad rule permits the Russell construction and exposes the contradiction.
What does typed mode mean?
Typed mode separates objects by levels. It rejects self membership before the paradox forms, so the dangerous expression is blocked.
What is restricted class mode?
Restricted class mode treats dangerous universal collections as non ordinary sets. The rule may describe a class, but not a memberable set.
Can I export results?
Yes. After calculation, use the CSV button for spreadsheet data or the PDF button for a simple printable report.
Who can use this page?
Students, teachers, writers, and logic learners can use it. It is best for quick demonstrations and classroom explanations.