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A full-fledged textbook that explains all the core results, techniques, and ideas in basic mathematics without assuming any special prior knowledge. This book explains through complete proofs how the ideas used to obtain results such as the ``Incompleteness Theorem'' and ``Proof of Independence of the Continuum Hypothesis.'' Includes practice questions and detailed answers for deeper understanding.

Message from the author

　It is difficult to explain the field of basic mathematics in one word, as is the case with any field. The reason for this is not only due to domestic circumstances, but also due to its unique position. This is because the theory of basic mathematics was born out of a critical awareness of the problem of ``mathematics about mathematics.''

　Many of the issues in basic mathematics theory have been rooted in mathematics for a long time. Fundamentals of mathematics clarified these and made them new subjects of research. The basic attitude of the basic theory of mathematics is to reflect on oneself (and others). When doing mathematics, we often ask ourselves questions such as ``What is mathematics doing?'' or ``What was I thinking about and how was I thinking about it?''

　This attitude easily leads one to fundamental questions such as ``computation,'' ``sets,'' ``proof,'' and ``definability.'' The question of what the basic concepts of mathematics are was the creation of the Theory of Fundamentals of Mathematics.

　Some of our readers may have heard terms such as the ``incompleteness theorem'' and ``independence proof of the continuum hypothesis.'' The former can be thought of as a theorem regarding the relationship between the irrefutable fact that ``proof (the act of) is a kind of symbolic calculation that can be confirmed by mechanical calculation'' and ``the correctness of a proposition.'' The latter is one answer to the question that anyone may have: ``What is the size (cardinality) of the set of all real numbers?'' In this book, I will explain through proofs the kind of thinking that led to the results regarding such a fundamental problem. At the same time, I aim to provide a panoramic view of the field of basic mathematics in one book.

　Fundamental theory is a relatively young field, having been around for about 100 years. Research on the basic theory of mathematics continues to deepen and expand at an extremely rapid pace. At present, it is far beyond the author's ability to look at the entire picture, not only because of the space available. Here we will try to introduce only the basics.

--From the "Foreword" of this book