- Enriching the basic language(what Internal Set Theory does to mathematics)
- Adding new words, symbols, or concepts to the vocabulary we use in mathematics so we can talk about and work with more things.
- Internal Set Theory(the main subject of the statement)
- A mathematical framework created by Abraham Robinson that treats infinite and infinitesimal (incredibly tiny) numbers as legitimate objects you can actually work with, rather than just useful shortcuts.
- Non-standard reals(contrasted with standard reals)
- Numbers that exist in extended number systems but aren't part of the usual real numbers; they include infinitesimals (numbers infinitely close to zero) and infinite numbers that standard math doesn't normally use.
- Standard reals(contrasted with non-standard reals)
- The ordinary real numbers you learn about in regular math class—numbers like 1, 2.5, π, and √2 that can be located precisely on a number line.
- asymmetry(Modal logic frame semantics)
- A frame property expressible in hybrid logic by the formula c→□¬◇c, meaning if world x accesses world y, then y does not access x.
- external sets(Internal Set Theory; examples include the set of all infinitesimals and the set of all standard real numbers)
- Sets that cannot be defined within the basic language of Internal Set Theory and do not necessarily satisfy properties like the least upper bound property.
- infinitesimals(Peirce's philosophy of mathematics and foundations of calculus)
- Quantities that constitute the 'glue' causing points on a continuous line to lose their individual identity, thereby grounding the concept of a true continuum
- internal sets(Internal Set Theory)
- Sets that can be defined in the basic language of Internal Set Theory; they behave the same as standard sets of standard reals, including the property that bounded internal sets always have a least upper bound.